## string theory, quantum cosmology, etc.

School and Workshop on Modular Forms and Black Holes, 5–14 January 2017, National Institute of Science Education and Research, Bhubaneswar, India

E. Elizalde, S. Leseduarte and S. Zerbini, "Mellin transform techniques for zeta-function resummations"

"Making use of inverse Mellin transform techniques for analytical continuation, an elegant proof and an extension of the zeta function regularization theorem is obtained...As an application of the method, the summation of the series which appear in the analytic computation (for different ranges of temperature) of the partition function of the string - basic in order to ascertain if QCD is some limit of a string theory - is performed. "

E. Elizalde, S. Leseduarte and S.D. Odintsov, "Partition functions for the rigid string and membrane at any temperature"

"Exact expressions for the partition functions of the rigid string and membrane at any temperature are obtained in terms of hypergeometric functions. By using zeta function regularization methods, the results are analytically continued and written as asymptotic sums of Riemann-Hurwitz zeta functions, which provide very good numerical approximations with just a few first terms."

A class of zeta functions that extends the class of Epstein's was recently brought to my attention by Prof. E. Elizalde of M.I.T. They are spectral zeta functions associated with a quadratic + linear + constant form in any number of dimensions. Elizalde has developed formulas for them which extend the famous Chowla-Selberg formula.

E. Elizalde, "Explicit zeta functions for bosonic and fermionic fields on a noncommutative toroidal spacetime", Journal of Physics A 34 (2001) 3025–3036

E. Elizalde, "Multidimensional extension of the generalized Chowla-Selberg formula", Communications in Mathematical Physics 198 (1998) 83–95

E. Elizalde, "Zeta functions, formulas and applications", J. Comp. Appl. Math. 118 (2000) 125

G.W. Moore, "Les Houches lectures on strings and arithmetic" (preprint 01/04)

[abstract:] "These are lecture notes for two lectures delivered at the Les Houches workshop on Number Theory, Physics, and Geometry, March 2003. They review two examples of interesting interactions between number theory and string compactification, and raise some new questions and issues in the context of those examples. The first example concerns the role of the Rademacher expansion of coefficients of modular forms in the AdS/CFT correspondence. The second example concerns the role of the "attractor mechanism" of supergravity in selecting certain arithmetic Calabi-Yau's as distinguished compactifications."

B. Dragovich, "Nonlocal dynamics of $p$-adic strings" (preprint 11/2010)

[abstract:] "We consider the construction of Lagrangians that might be suitable for describing the entire $p$-adic sector of an adelic open scalar string. These Lagrangians are constructed using the Lagrangian for $p$-adic strings with an arbitrary prime number $p$. They contain space-time nonlocality because of the d'Alembertian in argument of the Riemann zeta function. We present a brief review and some new results."

B. Dragovich, "On p-adic sector of adelic string" (Presented at the 2nd Conference on SFT and Related Topics, Moscow, April 2009. Submitted to Theor. Math. Phys.)

[abstract:] "We consider construction of Lagrangians which are candidates for p-adic sector of an adelic open scalar string. Such Lagrangians have their origin in Lagrangian for a single p-adic string and contain the Riemann zeta function with the 'Alembertian in its argument. In particular, we present a new Lagrangian obtained by an additive approach which takes into account all p-adic Lagrangians. The very attractive feature of this new Lagrangian is that it is an analytic function of the d'Alembertian. Investigation of the field theory with Riemann zeta function is interesting in itself as well."

B. Dragovich, "Towards effective Lagrangians for adelic strings" (preprint 02/2009)

B. Dragovich, "Some Lagrangians with zeta function nonlocality" (preprint, 05/2008)

[abstract:] "Some nonlocal and nonpolynomial scalar field models originated from $p$-adic string theory are considered. Infinite number of spacetime derivatives is governed by the Riemann zeta function through d'Alembertian $\Box$ in its argument. Construction of the corresponding Lagrangians begins with the exact Lagrangian for effective field of $p$-adic tachyon string, which is generalized replacing $p$ by arbitrary natural number $n$ and then taken a sum of over all $n$. Some basic classical field properties of these scalar fields are obtained. In particular, some trivial solutions of the equations of motion and their tachyon spectra are presented. Field theory with Riemann zeta function nonlocality is also interesting in its own right."

B. Dragovich, "Zeta nonlocal scalar fields" (preprint, 04/2008)

[abstract:] "We consider some nonlocal and nonpolynomial scalar field models originated from p-adic string theory. Infinite number of spacetime derivatives is determined by the operator valued Riemann zeta function through d'Alembertian $\Box$ in its argument. Construction of the corresponding Lagrangians L starts with the exact Lagrangian $\mathcal{L}_p$ for effective field of p-adic tachyon string, which is generalized replacing p by arbitrary natural number n and then taken a sum of $\mathcal{L}_n$ over all n. The corresponding new objects we call zeta scalar strings. Some basic classical field properties of these fields are obtained and presented in this paper. In particular, some solutions of the equations of motion and their tachyon spectra are studied. Field theory with Riemann zeta function dynamics is interesting in its own right as well."

B. Dragovich, "Zeta strings" (preprint 03/2007)

[abstract:] "We introduce nonlinear scalar field models for open and open-closed strings with spacetime derivatives encoded in the operator valued Riemann zeta function. The corresponding two Lagrangians are derived in an adelic approach starting from the exact Lagrangians for effective fields of $p$-adic tachyon strings. As a result tachyons are absent in these models. These new strings we propose to call zeta strings. Some basic classical properties of the zeta strings are obtained and presented in this paper."

B. Dragovich, "Lagrangians with Riemann zeta function" (preprint 08/2008)

[abstract:] "We consider construction of some Lagrangians which contain the Riemann zeta function. The starting point in their construction is $p$-adic string theory. These Lagrangians describe some nonlocal and nonpolynomial scalar field models, where nonlocality is controlled by the operator valued Riemann zeta function. The main motivation for this research is intention to find an effective Lagrangian for adelic scalar strings."

R. Auzzi and S. Elitzur and S.B. Gudnason and E. Rabinovici, "Time-dependent stabilization in AdS/CFT" (preprint 06/2012)

[abstract:] "We consider theories with time-dependent Hamiltonians which alternate between being bounded and unbounded from below. For appropriate frequencies dynamical stabilization can occur rendering the effective potential of the system stable. We first study a free field theory on a torus with a time-dependent mass term, finding that the stability regions are described in terms of the phase diagram of the Mathieu equation. Using number theory we have found a compactification scheme such as to avoid resonances for all momentum modes in the theory. We further consider the gravity dual of a conformal field theory on a sphere in three spacetime dimensions, deformed by a doubletrace operator. The gravity dual of the theory with a constant unbounded potential develops big crunch singularities; we study when such singularities can be cured by dynamical stabilization. We numerically solve the Einstein-scalar equations of motion in the case of a time-dependent doubletrace deformation and find that for sufficiently high frequencies the theory is dynamically stabilized and big crunches get screened by black hole horizons."

C. Angelantonj, M. Cardella, S. Elitzur, E. Rabinovici, "Vacuum stability, string density of states and the Riemann zeta function" (preprint 12/2010)

[abstract:] "We study the distribution of graded degrees of freedom in classically stable oriented closed string vacua and use the Rankin-Selberg transform to link it to the finite one-loop vacuum energy. In particular, we find that the spectrum of physical excitations not only must enjoy asymptotic supersymmetry but actually, at very large mass, bosonic and fermionic states must follow a universal oscillating pattern, whose frequencies are related to the zeros of the Riemann zeta-function. Moreover, the convergence rate of the overall number of the graded degrees of freedom to the value of the vacuum energy is determined by the Riemann hypothesis. We discuss also attempts to obtain constraints in the case of tachyon-free open-string theories."

M.A. Cardella, "Error estimates in horocycle averages asymptotics: Challenges from string theory" (preprint 12/2010)

[abstract:] "We study asymptotics and error estimates of long horocycle averages of automorphic functions with not-so-mild growing conditions at the cusp. For modular functions of rapid decay, it is a classical result that a certain value of the power in the error estimate is equivalent to the Riemann hypothesis. For modular functions of polynomial growth, we study how asymptotics are modified, by devising and unfolding trick with a two-dimensional lattice theta series. For modular functions of exponential growth, we gain insights on the horocycle average asymptotic, by translating this quantity into a states counting function in heterotic string theory. Consistency conditions for the heterotic string physical spectrum, lead to a special bound on the modular function exponential growth. String theory suggests that automorphic functions in the growing class below that bound should have convergent horocycle average."

S.L. Cacciatori and M. Cardella, "Equidistribution rates, genus $g$ closed string amplitudes, and the Riemann Hypothesis" (preprint 07/2010)

[abstract:] "Equidistribution of unipotent averages of $Sp(2g,\mathbb{R})$ automorphic forms is discussed by using analytic methods. We find that certain values of the equidistribution convergence rates are only compatible with the Riemann hypothesis. These results have applications in string theory by using modular functions appearing in recently proposed genus $g \ge 2$ closed string amplitudes. The potential mathematical advantages of obtaining a mapping of the equidistribution convergence rates into corresponding quantities appearing in string theory is also outlined."

M. McGuigan, "Riemann Hypothesis and short distance fermionic Green's functions" (preprint 04/05)

[abstract:] "We show that the Green's function of a two dimensional fermion with a modified dispersion relation and short distance parameter a is given by the Lerch zeta function. The Green's function is defined on a cylinder of radius R and we show that the condition R = a yields the Riemann zeta function as a quantum transition amplitude for the fermion. We formulate the Riemann hypothesis physically as a nonzero condition on the transition amplitude between two special states associated with the point of origin and a point half way around the cylinder each of which are fixed points of a $Z_2$ transformation. By studying partial sums we show that that the transition amplitude formulation is analogous to neutrino mixing in a low dimensional context. We also derive the thermal partition function of the fermionic theory and the thermal divergence at temperature 1/a. In an alternative harmonic oscillator formalism we discuss the relation to the fermionic description of two dimensional string theory and matrix models. Finally we derive various representations of the Green's function using energy momentum integrals, point particle path integrals, and string propagators."

M. McGuigan, "Riemann Hypothesis, matrix/gravity correspondence and FZZT brane partition functions" (preprint 08/2007)

[abstract:] "We investigate the physical interpretation of the Riemann zeta function as a FZZT brane partition function associated with a matrix/gravity correspondence. The Hilbert-Polya operator in this interpretation is the master matrix of the large N matrix model. Using a related function $\Xi(z)$ we develop an analogy between this function and the Airy function Ai(z) of the Gaussian matrix model. The analogy gives an intuitive physical reason why the zeros lie on a critical line. Using a Fourier transform of the $\Xi(z)$ function we identify a Kontsevich integrand. Generalizing this integrand to $n \times n$ matrices we develop a Kontsevich matrix model which describes n FZZT branes. The Kontsevich model associated with the $\Xi(z)$ function is given by a superposition of Liouville type matrix models that have been used to describe matrix model instantons."

M. McGuigan, "Riemann Hypothesis and master matrix for FZZT brane partition functions" (preprint 05/2008)

[abstract:] "We continue to investigate the physical interpretation of the Riemann zeta function as a FZZT brane partition function associated with a matrix/gravity correspondence begun in arxiv:0708.0645. We derive the master matrix of the $(2,1)$ minimal and $(3,1)$ minimal matrix model. We use it's characteristic polynomial to understand why the zeros of the FZZT partition function, which is the Airy function, lie on the real axis. We also introduce an iterative procedure that can describe the Riemann $\Xi$ function as a deformed minimal model whose deformation parameters are related to a Konsevich integrand. Finally we discuss the relation of our work to other approaches to the Riemann $\Xi$ function including expansion in terms of Meixner-Pollaczek polynomials and Riemann-Hilbert problems."

Yang-Hui He, V. Jejjala, D. 'Minic, "From Veneziano to Riemann: A string theory statement of the Riemann Hypothesis" (preprint 01/2015)

[abstract:] "We discuss a precise relation between the Veneziano amplitude of string theory, rewritten in terms of ratios of the Riemann zeta function, and two elementary criteria for the Riemann hypothesis formulated in terms of integrals of the logarithm and the argument of the zeta function. We also discuss how the integral criterion based on the argument of the Riemann zeta function relates to the Li criterion for the Riemann hypothesis. We provide a new generalization of this integral criterion. Finally, we comment on the physical interpretation of our recasting of the Riemann hypothesis in terms of the Veneziano amplitude."

Yang-Hui He, V. Jejjala, D. Minic, "On the physics of the Riemann zeros" (Quantum Theory and Symmetries 6 conference proceedings)

[abstract:] "We discuss a formal derivation of an integral expression for the Li coefficients associated with the Riemann xi-function which, in particular, indicates that their positivity criterion is obeyed, whereby entailing the criticality of the non-trivial zeros. We conjecture the validity of this and related expressions without the need for the Riemann Hypothesis and discuss a physical interpretation of this result within the Hilbert-Polya approach. In this context we also outline a relation between string theory and the Riemann Hypothesis."

Jen-Chi Lee, Yi Yang, Sheng-Lan Ko, "Stirling number Identities and high energy string scatterings" (Invited talk presented by Jen-Chi Lee at "Tenth Workshop on QCD", Paris, France, June 7-12, 2009. To be published in the SLAC eConf series)

[abstract] "We use Stirling number identities developed recently in number theory to show that ratios among high energy string scattering amplitudes in the fixed angle regime can be extracted from the Kummer function of the second kind. This result not only brings an interesting bridge between string theory and combinatoric number theory but also sheds light on the understanding of algebraic structure of high energy stringy symmetry."

P. Ranjan Giri and R.K. Bhaduri, "Physical interpretation for Riemann zeros from black hole physics" (preprint 05/2009)

[abstract:] "According to a conjecture attributed to Poyla and Hilbert, there is a self-adjoint operator whose eigenvalues are the the nontrivial zeros of the Riemann zeta function. We show that the near-horizon dynamics of a massive scalar field in the Schwarzschild black hole spacetime, under a reasonable boundary condition, gives rise to energy eigenvalues that coincide with the Riemann zeros. In achieving this result, we exploit the Bekenstein conjecture of black hole area quantization, and argue that it is responsible for the breaking of the continuous scale symmetry of the near horizon dynamics into a discrete one."

[abstract:] "In this article we present the post-Newtonian (pN) coefficients of the energy flux (and angular momentum flux) at infinity and event horizon for a particle in circular, equatorial orbits about a Kerr black hole (of mass $M$ and spin-parameter $a$) up to 20-pN order. When a pN term is not a polynomial in $a/M$ and includes irrational functions (like polygamma functions), it is written as a power series of $a/M$. This is achieved by calculating the fluxes numerically with an accuracy greater than 1 part in 10600. Such high accuracy allows us to extract analytical values of pN coefficients that are linear combinations of transcendentals like the Euler constant, logarithms of prime numbers and powers of $\pi$. We also present the 20-pN expansion (spin-independent pN expansion) of the ingoing energy flux at the event horizon for a particle in circular orbit about a Schwarzschild black hole.

E. Belbruno, "On the regularizability of the Big Bang singularity" (preprint 05/2012)

[abstract:] "The singularity for the big bang state can be represented using the generalized anisotropic Friedmann equation, resulting in a system of differential equations in a central force field. We study the regularizability of this singularity as a function of a parameter, the state variable, $w$. We prove that for $w >1$ it is regularizable only for $w$ satisfying relative prime number conditions, and for $w \leq 1$ it can always be regularized. This is done by using a McGehee transformation, usually applied in the three and four-body problems. This transformation blows up the singularity into an invariant manifold. This has implications on the idea that our universe could have resulted from a big crunch of a previous universe, assuming a Friedmann modeling."

J. Wang, "The zeros and poles of the partition function" (preprint 03/2011)

[abstract:] "In this paper, we consider the physical meaning of the zeros and poles of partition function. We consider three different systems, including the harmonic oscillator in one dimension, Riemann zeta function and the quasinormal modes of black hole."

J. Manuel Garcia-Islas, Black hole entropy in loop quantum gravity and number theory (preprint 07/09)

[abstract:] "We show that counting different configurations that give rise to black hole entropy in loop quantum gravity is related to partitions in number theory."

A. Dabholkar, J. Gomes and S. Murthy, "Nonperturbative black hole entropy and Kloosterman sums" (preprint 03/2014)

[abstract:] "Non-perturbative quantum corrections to supersymmetric black hole entropy often involve nontrivial number-theoretic phases called Kloosterman sums. We show how these sums can be obtained naturally from the functional integral of supergravity in asymptotically AdS_2 space for a class of black holes. They are essentially topological in origin and correspond to charge-dependent phases arising from the various gauge and gravitational Chern–Simons terms and boundary Wilson lines evaluated on Dehn-filled solid 2-torus. These corrections are essential to obtain an integer from supergravity in agreement with the quantum degeneracies, and reveal an intriguing connection between topology, number theory, and quantum gravity. We give an assessment of the current understanding of quantum entropy of black holes."

A. Corichi, "Black holes and entropy in loop quantum gravity: An overview" (preprint 01/2009)

[abstract:] "Black holes in equilibrium and the counting of their entropy within Loop Quantum Gravity are reviewed. In particular, we focus on the conceptual setting of the formalism, briefly summarizing the main results of the classical formalism and its quantization. We then focus on recent results for small, Planck scale, black holes, where new structures have been shown to arise, in particular an effective quantization of the entropy. We discuss recent results that employ in a very effective manner results from number theory, providing a complete solution to the counting of black hole entropy. We end with some comments on other approaches that are motivated by loop quantum gravity."

M. Axenides, E. Floratos and S. Nicolis, "Chaotic information processing by extremal black holes" (preprint 04/2015)

[abstract:] "We review an explicit regularization of the AdS2/CFT1 correspondence, that preserves all isometries of bulk and boundary degrees of freedom. This scheme is useful to characterize the space of the unitary evolution operators that describe the dynamics of the microstates of extremal black holes in four spacetime dimensions. Using techniques from algebraic number theory to evaluate the transition amplitudes, we remark that the regularization scheme expresses the fast quantum computation capability of black holes as well as its chaotic nature."

H.C. Rosu and M. Planat, "On arithmetic detection of grey pulses with application to Hawking radiation" (preprint 05/2002)

[abstract:] "Micron-sized black holes do not necessarily have a constant horizon temperature distribution. The black hole remote-sensing problem means to find out the surface' temperature distribution of a small black hole from the spectral measurement of its (Hawking) grey pulse. This problem has been previously considered by Rosu, who used Chen's modified Möbius inverse transform. Here, we hint on a Ramanujan generalization of Chen's modified Möbius inverse transform that may be considered as a special wavelet processing of the remote-sensed grey signal coming from a black hole or any other distant grey source."

H.C. Rosu, "Quantum Hamiltonians and prime numbers", Modern Physics Letters A 18 (2003)

[abstract:] "A short review of Schroedinger hamiltonians for which the spectral problem has been related in the literature to the distribution of the prime numbers is presented here. We notice a possible connection between prime numbers and centrifugal inversions in black holes and suggest that this remarkable link could be directly studied within trapped Bose-Einstein condensates. In addition, when referring to the factorizing operators of Pitkanen and Castro and collaborators, we perform a mathematical extension allowing a more standard supersymmetric approach."

This very welcome, thorough review article discusses and compares the various inter-related work of Bhaduri-Khare-Law, Berry-Keating, Aneva, Castro, et.al., Pitkanen, Khuri, Joffily, Wu-Sprung, Okubo, Mussardo, Boos-Korepin, Crehan and others.

A. Sugamoto, "Factorization of number into prime numbers viewed as decay of particle into elementary particles conserving energy" (preprint 10/2009)

[abstract:] "Number theory is considered, by proposing quantum mechanical models and string-like models at zero and finite temperatures, where the factorization of number into prime numbers is viewed as the decay of particle into elementary particles conserving energy. In these models, energy of a particle labeled by an integer $n$ is assumed or derived to being proportional to $\ln n$. The one-loop vacuum amplitudes, the free energies and the partition functions at finite temperature of the string-like models are estimated and compared with the zeta functions. The $SL(2, {\bf Z})$ modular symmetry, being manifest in the free energies is broken down to the additive symmetry of integers, ${\bf Z}_{+}$, after interactions are turned on. In the dynamical model existing behind the zeta function, prepared are the fields labeled by prime numbers. On the other hand the fields in our models are labeled, not by prime numbers but by integers. Nevertheless, we can understand whether a number is prime or not prime by the decay rate, namely by the corresponding particle can decay or can not decay through interactions conserving energy. Among the models proposed, the supersymmetric string-like model has the merit of that the zero point energies are cancelled and the energy levels may be stable against radiative corrections."

M. Cvetič, I. García-Etxebarria, J. Halverson, "On the computation of non-perturbative effective potentials in the string theory landscape -- IIB/F-theory perspective" (preprint 09/2010)

[abstract:] "We discuss a number of issues arising when computing non-perturbative effects systematically across the string theory landscape. In particular, we cast the study of fairly generic physical properties into the language of computability/number theory and show that this amounts to solving systems of diophantine equations. In analogy to the negative solution to Hilbert's 10th problem, we argue that in such systematic studies there may be no algorithm by which one can determine all physical effects. We take large volume type IIB compactifications as an example, with the physical property of interest being the low-energy non-perturbative F-terms of a generic compactification. A similar analysis is expected to hold for other kinds of string vacua, and we discuss in particular the extension of our ideas to F-theory. While these results imply that it may not be possible to answer systematically certain physical questions about generic type IIB compactifications, we identify particular Calabi-Yau manifolds in which the diophantine equations become linear, and thus can be systematically solved. As part of the study of the required systematics of F-terms, we develop technology for computing Z_2 equivariant line bundle cohomology on toric varieties, which determines the presence of particular instanton zero modes via the Koszul complex. This is of general interest for realistic IIB model building on complete intersections in toric ambient spaces."

S. Davis, "Spin structures on Riemann surfaces and the perfect numbers" (preprint 12/1998)

"The equality between the number of odd spin structures on a Riemann surface of genus $g$, with $2^g - 1$ being a Mersenne prime, and the even perfect numbers, is an indication that the action of the modular group on the set of spin structures has special properties related to the sequence of perfect numbers. A method for determining whether Mersenne numbers are primes is developed by using a geometrical representation of these numbers. The connection between the non-existence of finite odd perfect numbers and the irrationality of the square root of twice the product of a sequence of repunits is investigated, and it is demonstrated, for an arbitrary number of prime factors, that the products of the corresponding repunits will not equal twice the square of a rational number."

Related work by S. Davis:

"A method for generating Mersenne primes and the extent of the sequence of even perfect numbers" (preprint, again involving spin structures and dynamical systems)

"A rationality condition for the existence of odd perfect numbers" (preprint 11/2000)

"A proof of the odd perfect number conjecture" (preprint 01/2004)

P. Frampton and T. Kephart, "Mersenne primes, polygonal anomalies and string theory classification"

"It is pointed out that the Mersenne primes Mp = 2p-1 and associated perfect numbers Mp = 2p-1Mp play a significant role in string theory; this observation may suggest a classification of consistent string theories."

P.H. Frampton and Y. Okada, "p-Adic string N-point function", Phys. Rev. Lett. B 60 (1988) 484–486

N. Efremov and N.V. Mitskievich, "A T0-discrete universe model with five low-energy fundamental interactions and the coupling constants hierarchy" (preprint, 09/03)

[abstract:] "A quantum model of universe is constructed in which values of dimensionless coupling constants of the fundamental interactions (including the cosmological constant) are determined via certain topological invariants of manifolds forming finite ensembles of 3D Seifert fibrations. The characteristic values of the coupling constants are explicitly calculated as the set of rational numbers (up to the factor 2) on the basis of a hypothesis that these values are proportional to the mean relative fluctuations of discrete volumes of manifolds in these ensembles. The discrete volumes are calculated using the standard Alexandroff procedure of constructing T0-discrete spaces realized as nerves corresponding to characteristic canonical triangulations which are compatible with the Milnor representation of Seifert fibered homology spheres being the building material of all used 3D manifolds. Moreover, the determination of all involved homology spheres is based on the first nine prime numbers (p1 =2, ..., p9=23). The obtained hierarchy of coupling constants at the present evolution stage of universe well reproduces the actual hierarchy of the experimentally observed dimensionless low-energy coupling constants."

D.B. Grunberg, "Integrality of open instanton numbers" (preprint 05/03)

[abstract:] "We prove the integrality of the open instanton numbers in two examples of counting holomorphic disks on local Calabi-Yau threefolds: the resolved conifold and the degenerate P x P. Given the B-model superpotential, we extract by hand all Gromow-Witten invariants in the expansion of the A-model superpotential. The proof of their integrality relies on enticing congruences of binomial coefficients modulo powers of a prime. We also derive an expression for the factorial (pk-1)! modulo powers of the prime p. We generalise two theorems of elementary number theory, by Wolstenholme and by Wilson."

P. D'Eath and G. Esposito, "The effect of boundaries in one-loop quantum cosmology"

P. D'Eath and G. Esposito, "Local Boundary Conditions for the Dirac Operator and One-Loop Quantum Cosmology"

P. D'Eath and G. Esposito, "Spectral boundary conditions in one-loop quantum cosmology"

"For fermionic fields on a compact Riemannian manifold with boundary one has a choice between local and non-local (spectral) boundary conditions. The one-loop prefactor in the Hartle-Hawking amplitude in quantum cosmology can then be studied using the generalized Riemann zeta-function formed from the squared eigenvalues of the four-dimensional fermionic operators."

A.P. de Almeida, F.T. Brandt and J. Frenkel, "Thermal matter and radiation in a gravitational field"

"We study the one-loop contributions of matter and radiation to the gravitational polarization tensor at finite temperatures. Using the analytically continued imaginary-time formalism, the contribution of matter is explicitly given to next-to-leading T2 order. We obtain an exact form for the contribution of radiation fields, expressed in terms of generalized Riemann zeta functions."

V. V. Nesterenko and I. G. Pirozhenko, "Justification of the zeta function renormalization in rigid string model"

"A consistent procedure for regularization of divergences and for the subsequent renormalization of the string tension is proposed in the framework of the one-loop calculation of the interquark potential generated by the Polyakov-Kleinert string. In this way, a justification of the formal treatment of divergences by analytic continuation of the Riemann and Epstein-Hurwitz zeta functions is given. A spectral representation for the renormalized string energy at zero temperature is derived, which enables one to find the Casimir energy in this string model at nonzero temperature very easy."

A. Edery, "Multidimensional cut-off technique, odd-dimensional Epstein zeta functions and Casimir energy of massless scalar fields", submitted to J. Physics A

[abstract:] "Quantum fluctuations of massless scalar fields represented by quantum fluctuations of the quasiparticle vacuum in a zero-temperature dilute Bose-Einstein condensate may well provide the first experimental arena for measuring the Casimir force of a field other than the electromagnetic field. This would constitute a real Casimir force measurement - due to quantum fluctuations - in contrast to thermal fluctuation effects. We develop a multidimensional cut-off technique for calculating the Casimir energy of massless scalar fields in d-dimensional rectangular spaces with q large dimensions and d-q dimensions of length L and generalize the technique to arbitrary lengths. We explicitly evaluate the multidimensional remainder and express it in a form that converges exponentially fast. Together with the compact analytical formulas we derive, the numerical results are exact and easy to obtain. Most importantly, we show that the division between analytical and remainder is not arbitrary but has a natural physical interpretation. The analytical part can be viewed as the sum of individual parallel plate energies and the remainder as an interaction energy. In a separate procedure, via results from number theory, we express some odd-dimensional homogeneous Epstein zeta functions as products of one-dimensional sums plus a tiny remainder and calculate from them the Casimir energy via zeta function regularization."

V. Di Clemente, S. F. King and D.A.J. Rayner, "Supersymmetry and electroweak breaking with large and small extra dimensions", Nucl. Phys. B 617 (2001) 71–100

[abstract:] "We consider the problem of supersymmetry and electroweak breaking in a 5d theory compactified on an $S^{1}/Z_{2}$ orbifold, where the extra dimension may be large or small. We consider the case of a supersymmetry breaking 4d brane located at one of the orbifold fixed points with the Standard Model gauge sector, third family and Higgs fields in the 5d bulk, and the first two families on a parallel 4d matter brane located at the other fixed point. We compute the Kaluza-Klein mass spectrum in this theory using a matrix technique which allows us to interpolate between large and small extra dimensions. We also consider the problem of electroweak symmetry breaking in this theory and localize the Yukawa couplings on the 4d matter brane spatially separated from the brane where supersymmetry is broken. We calculate the 1-loop effective potential using a zeta-function regularization technique, and find that the dominant top and stop contributions are separately finite. Using this result we find consistent electroweak symmetry breaking for a compactification scale {$1/R \approx 830$ GeV} and a lightest Higgs boson mass $m_{h} \approx 170$ GeV."

K. Roland, "Two- and three-loop amplitudes in covariant loop calculus", Nuclear Physics B 313 (1989) 432–446

[abstract:] "We study two- and three-loop vacuum amplitudes for the closed bosonic string. We compare two sets of expressions for the corresponding density on moduli space. One is based on the covariant reggeon loop calculus (where modular invariance is not manifest). The other is based on analytic geometry. We want to prove identity between the two sets of expressions. Quite apart from demonstrating modular invariance of the reggeon results, this would in itself be a remarkable mathematical feature. Identity is established to 'high' order in some moduli and exactly in others. The expansions reveal an essentially number-theoretic structure. Agreement is found only by exploiting the connection between the four Jacobi theta-functions and number theory."

J. L. Petersen, K. O. Roland and J. R. Sidenius, "Modular invariance and covariant loop calculus", Physics Letters B 205 (1988) 262–266

[abstract:] "The covariant loop calculus provides an efficient technique for computing explicit expressions for the density on moduli space corresponding to arbitrary (bosonic string) loop diagrams. Since modular invariance is not manifest, however, we carry out a detailed comparison with known explicit two- and three-loop results derived using analytic geometry (one loop is known to be okay). We establish identity to 'high' order in some moduli and exactly in others. Agreement is found as a result of various non-trivial cancellations, in part related to number theory. We feel our results provide very strong support for the correctness of the covariant loop calculus approach."

G. Heinrich, T. Huber, D. Maitre, "Master integrals for fermionic contributions to massless three-loop form factors" (preprint 12/07)

[abstract:] "In this letter we continue the calculation of master integrals for massless three-loop form factors by giving analytical results for those diagrams which are relevant for the fermionic contributions proportional to N_F^2, N_F*N, and N_F/N. Working in dimensional regularisation, we express one of the diagrams in a closed form which is exact to all orders in epsilon, containing Gamma-functions and hypergeometric functions of unit argument. In all other cases we derive multiple Mellin-Barnes representations from which the coefficients of the Laurent expansion in epsilon are extracted in an analytical form. To obtain the finite part of the three-loop quark and gluon form factors, all coefficients through transcendentality six in the Riemann zeta-function have to be included."

G. Heinrich, T. Huber, D. A. Kosower, V. A. Smirnov, "Nine-propagator master integrals for massless three-loop form factors" (preprint 02/2009)

[abstract:] "We complete the calculation of master integrals for massless three-loop form factors by computing the previously-unknown three diagrams with nine propagators in dimensional regularisation. Each of the integrals yields a six-fold Mellin-Barnes representation which we use to compute the coefficients of the Laurent expansion in epsilon. Using Riemann zeta functions of up to weight six, we give fully analytic results for one integral; for a second, analytic results for all but the finite term; for the third, analytic results for all but the last two coefficients in the Laurent expansion. The remaining coefficients are given numerically to sufficiently high accuracy for phenomenological applications."

L.J. Dixon, J.M. Drummond, M. von Hippel and J. Pennington, "Hexagon functions and the three-loop remainder function" (preprint 08/2013)

[abstract:] "We present the three-loop remainder function, which describes the scattering of six gluons in the maximally-helicity-violating configuration in planar $N=4$ super-Yang-Mills theory, as a function of the three dual conformal cross ratios. The result can be expressed in terms of multiple Goncharov polylogarithms. We also employ a more restricted class of "hexagon functions" which have the correct branch cuts and certain other restrictions on their symbols. We classify all the hexagon functions through transcendental weight five, using the coproduct for their Hopf algebra iteratively, which amounts to a set of first-order differential equations. The three-loop remainder function is a particular weight-six hexagon function, whose symbol was determined previously. The differential equations can be integrated numerically for generic values of the cross ratios, or analytically in certain kinematics limits, including the near-collinear and multi-Regge limits. These limits allow us to impose constraints from the operator product expansion and multi-Regge factorization directly at the function level, and thereby to fix uniquely a set of Riemann-zeta-valued constants that could not be fixed at the level of the symbol. The near-collinear limits agree precisely with recent predictions by Basso, Sever and Vieira based on integrability. The multi-Regge limits agree with the factorization formula of Fadin and Lipatov, and determine three constants entering the impact factor at this order. We plot the three-loop remainder function for various slices of the Euclidean region of positive cross ratios, and compare it to the two-loop one. For large ranges of the cross ratios, the ratio of the three-loop to the two-loop remainder function is relatively constant, and close to $-7$."

M.B. Green, J.G. Russo, P. Vanhove, "Low energy expansion of the four-particle genus-one amplitude in type II superstring theory" (preprint 01/2008)

[abstract:] "A diagrammatic expansion of coefficients in the low-momentum expansion of the genus-one four-particle amplitude in type II superstring theory is developed. This is applied to determine coefficients up to order $s^6R^4$ (where $s$ is a Mandelstam invariant and $R^4$ the linearized super-curvature), and partial results are obtained beyond that order. This involves integrating powers of the scalar propagator on a toroidal world-sheet, as well as integrating over the modulus of the torus. At any given order in $s$ the coefficients of these terms are given by rational numbers multiplying multiple zeta values (or Euler-Zagier sums) that, up to the order studied here, reduce to products of Riemann zeta values. We are careful to disentangle the analytic pieces from logarithmic threshold terms, which involves a discussion of the conditions imposed by unitarity. We further consider the compactification of the amplitude on a circle of radius $r$, which results in a plethora of terms that are power-behaved in $r$. These coefficients provide boundary data' that must be matched by any non-perturbative expression for the low-energy expansion of the four-graviton amplitude.

The paper includes an appendix by Don Zagier."

M.B. Green, S.D. Miller, J.G. Russo and P. Vanhove, "Eisenstein series for higher-rank groups and string theory amplitudes" (preprint 11/2011)

[abstract:] "Scattering amplitudes of superstring theory are strongly constrained by the requirement that they be invariant under dualities generated by discrete subgroups, $E_n(Z)$, of simply-laced Lie groups in the $E_n$ series ($n\leq 8$). In particular, expanding the four-supergraviton amplitude at low energy gives a series of higher derivative corrections to Einstein's theory, with coefficients that are automorphic functions with a rich dependence on the moduli. Boundary conditions supplied by string and supergravity perturbation theory, together with a chain of relations between successive groups in the $E_n$ series, constrain the constant terms of these coefficients in three distinct parabolic subgroups. Using this information we are able to determine the expressions for the first two higher derivative interactions (which are BPS-protected) in terms of specific Eisenstein series. Further, we determine key features of the coefficient of the third term in the low energy expansion of the four-supergraviton amplitude (which is also BPS-protected) in the $E_8$ case. This is an automorphic function that satisfies an inhomogeneous Laplace equation and has constant terms in certain parabolic subgroups that contain information about all the preceding terms."

E. D'Hoker and M.B. Green, "Zhang–Kawazumi invariants and superstring amplitudes" (preprint 08/2013)

[abstract:] "The two-loop Type II superstring correction to supergravity at order $D^6 R^4$ is derived from the genus-two superstring 4-point function of massless NS-NS states. We show that this correction is proportional to the integral over moduli space of a modular invariant introduced recently by Zhang and Kawazumi in number theory and related to the Faltings delta-invariant studied for genus-two by Bost. Furthermore, the structure of two-loop superstring corrections at higher order in the low energy expansion leads to higher invariants, which naturally generalize Zhang-Kawazumi invariant at genus two. An explicit formula is derived for the unique higher invariant associated with order $D^8 R^4$. In an attempt to compare the prediction for the $D^6 R^4$ correction from superstring perturbation theory with the one produced by S-duality and supersymmetry of Type IIB, various reformulations of the invariant are given. This comparison with string theory leads to a predicted value for the integral of the invariant over the moduli space of genus-two surfaces."

F. Brown, "Periods and Feynman amplitudes" (preprint 12/2015)

[abstract:] "Feynman amplitudes in perturbation theory form the basis for most predictions in particle collider experiments. The mathematical quantities which occur as amplitudes include values of the Riemann zeta function and relate to fundamental objects in number theory and algebraic geometry. This talk reviews some of the recent developments in this field, and explains how new ideas from algebraic geometry have led to much progress in our understanding of amplitudes. In particular, the idea that certain transcendental numbers, such as $\pi$, can be viewed as a representation of a group, provides a powerful framework to study amplitudes which reveals many hidden structures."

P. Fleig, H.P.A. Gustafsson, A. Kleinschmidt and D. Persson, "Eisenstein series and automorphic representations" (preprint 11/2015)

[abstract:] "We provide an introduction to the theory of Eisenstein series and automorphic forms on real simple Lie groups $G$, emphasising the role of representation theory. It is useful to take a slightly wider view and define all objects over the (rational) adeles $A$, thereby also paving the way for connections to number theory, representation theory and the Langlands program. Most of the results we present are already scattered throughout the mathematics literature but our exposition collects them together and is driven by examples. Many interesting aspects of these functions are hidden in their Fourier coefficients with respect to unipotent subgroups and a large part of our focus is to explain and derive general theorems on these Fourier expansions. Specifically, we give complete proofs of Langlands' constant term formula for Eisenstein series on adelic groups $G(A)$ as well as the Casselman--Shalika formula for the $p$-adic spherical Whittaker vector associated to unramified automorphic representations of $G(Q_p)$. Somewhat surprisingly, all these results have natural interpretations as encoding physical effects in string theory. We therefore introduce also some basic concepts of string theory, aimed toward mathematicians, emphasising the role of automorphic forms. In addition, we explain how the classical theory of Hecke operators fits into the modern theory of automorphic representations of adelic groups, thereby providing a connection with some key elements in the Langlands program, such as the Langlands dual group $LG$ and automorphic $L$-functions. Our treatise concludes with a detailed list of interesting open questions and pointers to additional topics where automorphic forms occur in string theory."

V. Ravindran, J. Smith and W.L. van Neerven, "Two-loop corrections to Higgs boson production" (Report-no: YITP-SB-04-46, 08/04)

[abstract:] "In this paper we present the two-loop vertex corrections to scalar and pseudo-scalar Higgs boson production for general colour factors for the gauge group SU(N) We derive a general formula for the vertex correction which holds for conserved and non conserved operators. For the conserved operator we take the electromagnetic vertex correction as an example whereas for the non conserved operators we take the two vertex corrections above. Our observations for the structure of the pole terms 1/4, 1/3 and 1/2 in two loop order are the same as made earlier in the literature for electromagnetism. For the single pole terms 1/ we can predict the terms containing the Riemann zeta functions (2) and (3).

S.K. Ashok, F. Cachazo, E. Dell'Aquila, "Children's drawings from Seiberg-Witten curves", Communications in Number Theory and Physics 1 no. 2 (2007) 237–305

[abstract:] "We consider $N=2$ supersymmetric gauge theories perturbed by tree level superpotential terms near isolated singular points in the Coulomb moduli space. We identify the Seiberg-Witten curve at these points with polynomial equations used to construct what Grothendieck called "dessins d'enfants" or "children's drawings" on the Riemann sphere. From a mathematical point of view, the dessins are important because the absolute Galois group $Gal(\bar{Q}/Q)$ acts faithfully on them. We argue that the relation between the dessins and Seiberg-Witten theory is useful because gauge theory criteria used to distinguish branches of $N=1$ vacua can lead to mathematical invariants that help to distinguish dessins belonging to different Galois orbits. For instance, we show that the confinement index defined in hep-th/0301006 is a Galois invariant. We further make some conjectures on the relation between Grothendieck's programme of classifying dessins into Galois orbits and the physics problem of classifying phases of $N=1$ gauge theories. "

S. Bose, J. Gundry and Y.-H. He, "Gauge theories and dessins d'enfants: Beyond the torus" (preprint /2014)

[abstract:] "Dessin d'enfants on elliptic curves are a powerful way of encoding doubly-periodic brane tilings, and thus, of four-dimensional supersymmetric gauge theories whose vacuum moduli space is toric, providing an interesting interplay between physics, geometry, combinatorics and number theory. We discuss and provide a partial classification of the situation in genera other than one by computing explicit Belyi pairs associated to the gauge theories. Important also is the role of the Igusa and Shioda invariants that generalise the elliptic $j$-invariant."

G. Moore, "Arithmetic and attractors" (preprint 07/03)

[abstract:] "We study relations between some topics in number theory and supersymmetric black holes. These relations are based on the "attractor mechanism" of N=2 supergravity. In IIB string compactification this mechanism singles out certain "attractor varieties". ' We show that these attractor varieties are constructed from products of elliptic curves with complex multiplication for N=4 and N=8 compactifications. The heterotic dual theories are related to rational conformal field theories. In the case of N=4 theories U-duality inequivalent backgrounds with the same horizon area are counted by the class number of a quadratic imaginary field. The attractor varieties are defined over fields closely related to class fields of the quadratic imaginary field. We discuss some extensions to more general Calabi-Yau compactifications and explore further connections to arithmetic including connections to Kronecker's Jugendtraum and the theory of modular heights. The paper also includes a short review of the attractor mechanism. A much shorter version of the paper summarizing the main points is the companion note entitled "Attractors and Arithmetic""

Yu. Manin and M. Marcolli, "Holography principle and arithmetic of algebraic curves", Adv. Theor. Math. Phys. 5 (2001), no. 3, 617–650.

[abstract:] "According to the holography principle (due to G. 't Hooft, L. Susskind, J. Maldacena, et al.), quantum gravity and string theory on certain manifolds with boundary can be studied in terms of a conformal field theory on the boundary. Only a few mathematically exact results corroborating this exciting program are known. In this paper we interpret from this perspective several constructions which arose initially in the arithmetic geometry of algebraic curves. We show that the relation between hyperbolic geometry and Arakelov geometry at arithmetic infinity involves exactly the same geometric data as the Euclidean AdS3 holography of black holes. Moreover, in the case of Euclidean AdS2 holography, we present some results on bulk/boundary correspondence where the boundary is a non-commutative space."

Yu. Manin, "Reflections on arithmetical physics", in Conformal Invariance and String Theory (Academic, 1989) 293–303

M. Marcolli's survey article "Number Theory in Physics" contains some material on string theory.

C. Hattori, M. Matsunaga, T. Matsuoka, K. Nakanishi, "Galois group on elliptic curves and flavor symmetry" (preprint 10/07)

[abstract:] "Putting emphasis on the relation between rational conformal field theory (RCFT) and algebraic number theory, we consider a brane configuration in which the D-brane intersection is an elliptic curve corresponding to RCFT. A new approach to the generation structure of fermions is proposed in which the flavor symmetry including the R-parity has its origin in the Galois group on elliptic curves with complex multiplication (CM). We study the possible types of the Galois group derived from the torsion points of the elliptic curve with CM. A phenomenologically viable example of the Galois group is presented, in which the characteristic texture of fermion masses and mixings is reproduced and the mixed-anomaly conditions are satisfied."

C. Castro, "On the Riemann Hypothesis and tachyons in dual string scattering amplitudes", International Journal of Geometric Methods in Modern Physics 3 no. 2 (2006) 187–199

[abstract:] "It is the purpose of this work to pursue a novel physical interpretation of the nontrivial Riemann zeta zeros and prove why the location of these zeros $z_n = 1/2+i\lambda_n$ corresponds physically to tachyonic-resonances/tachyonic-condensates, originating from the scattering of two on-shell tachyons in bosonic string theory. Namely, we prove that if there were nontrivial zeta zeros (violating the Riemann hypothesis) outside the critical line Real $z = 1/2$ (but inside the critical strip), these putative zeros do not correspond to any poles of the bosonic open string scattering (Veneziano) amplitude $A(s,t,u)$. The physical relevance of tachyonic-resonances/tachyonic-condensates in bosonic string theory, establishes an important connection between string theory and the Riemann Hypothesis. In addition, one has also a geometrical interpretation of the zeta zeros in the critical line in terms of very special (degenerate) triangular configurations in the upper-part of the complex plane."

"Supersymmetry, p-adic stochastic dynamics, Brownian motion, Fokker-Planck equation, Langevin equation, prime number random distribution, random matrices, p-adic fractal strings, the adelic condition, etc...are all deeply interconnected in this paper."

C. Bachas and I. Brunner, "Fusion of conformal interfaces" (preprint 12/2007)

[abstract:] "We study the fusion of conformal interfaces in the c=1 conformal field theory. We uncover an elegant structure reminiscent of that of black holes in supersymmetric theories. The role of the BPS black holes is played by topological interfaces, which (a) minimize the entropy function, (b) fix through an attractor mechanism one or both of the bulk radii, and (c) are (marginally) stable under splitting. One significant difference is that the conserved charges are logarithms of natural numbers, rather than vectors in a charge lattice, as for BPS states. Besides potential applications to condensed-matter physics and number theory, these results point to the existence of large solution-generating algebras in string theory."

"Here we present the results of applying the generalized Riemann zeta-function regularization method to the gravitational radiation reaction problem. We analyze in detail the headon collision of two nonspinning black holes with extreme mass ratio. The resulting reaction force on the smaller hole is repulsive. We discuss the possible extensions of these method to generic orbits and spinning black holes. The determination of corrected trajectories allows to add second perturbative corrections with the consequent increase in the accuracy of computed waveforms."

S. Benvenuti, B. Feng, A. Hanany, Yang-Hui He, "BPS operators in gauge theories: Quivers, syzygies and plethystics" (preprint 08/2006)

[abstract:] "We develop a systematic and efficient method of counting single-trace and multi-trace BPS operators for world-volume gauge theories of $N$ $D$-brane probes, for both $N \rightarrow \infty$ and finite $N$. The techniques are applicable to generic singularities, orbifold, toric, non-toric, et cetera, even to geometries whose precise field theory duals are not yet known. Mathematically, fascinating and intricate inter-relations between gauge theory, algebraic geometry, combinatorics and number theory exhibit themselves in the form of plethystics and syzygies."

Y.-H. He, "Graph zeta function and gauge theories" (preprint 02/2011)

[abstract:] "Along the recently trodden path of studying certain number theoretic properties of gauge theories, especially supersymmetric theories whose vacuum manifolds are non-trivial, we investigate Ihara's Graph Zeta Function for large classes of quiver theories and periodic tilings by bi-partite graphs. In particular, we examine issues such as the spectra of the adjacency and whether the gauge theory satisfies the strong and weak versions of the graph theoretical analogue of the Riemann Hypothesis."

[abstract:] "A Selberg zeta function is attached to the three-dimensional BTZ black hole, and a trace formula is developed for a general class of test functions. The trace formula differs from those of more standard use in physics in that the black hole has a fundamental domain of infinite hyperbolic volume. Various thermodynamic quantities associated with the black hole are conveniently expressed in terms of the zeta function."

"In the recent publication (Journal of Geometry and Physics, 33 (2000) 23-102) we demonstrated that dynamics of 2+1 gravity can be described in terms of train tracks. Train tracks were introduced by Thurston in connection with description of dynamics of surface automorphisms. In this work we provide an example of utilization of general formalism developed earlier. The complete exact solution of the model problem describing equilibrium dynamics of train tracks on the punctured torus is obtained. Being guided by similarities between the dynamics of 2d liquid crystals and 2+1 gravity the partition function for gravity is mapped into that for the Farey spin chain. The Farey spin chain partition function, fortunately, is known exactly and has been thoroughly investigated recently. Accordingly, the transition between the pseudo-Anosov and the periodic dynamic regime (in Thurston's terminology) in the case of gravity is being reinterpreted in terms of phase transitions in the Farey spin chain whose partition function is just a ratio of two Riemann zeta functions. The mapping into the spin chain is facilitated by recognition of a special role of the Alexander polynomial for knots/links in study of dynamics of self homeomorphisms of surfaces. At the end of paper, using some facts from the theory of arithmetic hyperbolic 3-manifolds (initiated by Bianchi in 1892), we develop systematic extension of the obtained results to noncompact Riemannian surfaces of higher genus. Some of the obtained results are also useful for 3+1 gravity. In particular, using the theorem of Margulis, we provide new reasons for the black hole existence in the Universe: black holes make our Universe arithmetic. That is the discrete Lie groups of motion are arithmetic."

"We study the fluctuation modes for lump solutions of the tachyon effective potential in p-adic open string theory. We find a discrete spectrum with equally spaced mass squared levels. We also find that the interactions derived from this field theory are consistent with p-adic string amplitudes for excited string

J.A. Nogueira, A. Maia, Jr., "Demonstration of how the zeta function method for effective potential removes the divergences"

[abstract:] "The calculation of the minimum of the effective potential using the zeta function method is extremely advantagous, because the zeta function is regular at s = 0 and we gain immediately a finite result for the effective potential without the necessity of subtraction of any pole or the addition of infinite counter-terms. The purpose of this paper is to explicitly point out how the cancellation of the divergences occurs and that the zeta function method implicitly uses the same procedure used by Bollini-Giambiagi and Salam-Strathdee in order to gain finite part of functions with a simple pole."

V.S. Vladimirov and Ya.I. Volovich, "On the nonlinear dynamical equation in the p-adic string theory" (preprint 06/03)

[abstract:] "In this work nonlinear pseudo-differential equations with the infinite number of derivatives are studied. These equations form a new class of equations which initially appeared in p-adic string theory. These equations are of much interest in mathematical physics and its applications in particular in string theory and cosmology. In the present work a systematical mathematical investigation of the properties of these equations is performed. The main theorem of uniqueness in some algebra of tempered distributions is proved. Boundary problems for bounded solutions are studied, the existence of a space-homogenous solution for odd p is proved. For even p it is proved that there is no continuous solutions and it is pointed to the possibility of existence of discontinuous solutions. Multidimensional equation is also considered and its soliton and q-brane solutions are discussed."

I. Ya. Aref'eva, I.V. Volovich, "Quantization of the Riemann zeta-function and cosmology" (preprint 12/2006)

[abstract:] "Quantization of the Riemann zeta-function is proposed. We treat the Riemann zeta-function as a symbol of a pseudodifferential operator and study the corresponding classical and quantum field theories. This approach is motivated by the theory of p-adic strings and by recent works on stringy cosmological models. We show that the Lagrangian for the zeta-function field is equivalent to the sum of the Klein-Gordon Lagrangians with masses defined by the zeros of the Riemann zeta-function. Quantization of the mathematics of Fermat-Wiles and the Langlands program is indicated. The Beilinson conjectures on the values of L-functions of motives are interpreted as dealing with the cosmological constant problem. Possible cosmological applications of the zeta-function field theory are discussed."

"It has been conjectured that an extremum of the tachyon potential of a bosonic D-brane represents the vacuum without any D-brane, and that various tachyonic lump solutions represent D-branes of lower dimension. We show that the tree level effective action of p-adic string theory, the expression for which is known exactly, provides an explicit realisation of these conjectures."

I.Ya. Aref'eva, M.G. Ivanov and I.V. Volovich, "Non-extremal intersecting p-branes in various dimensions", Phys. Lett. B 406 (1997) 44–48

[abstract:] "Non-extremal intersecting p-brane solutions of gravity coupled with several antisymmetric fields and dilatons in various space-time dimensions are constructed. The construction uses the same algebraic method of finding solutions as in the extremal case and a modified "no-force" conditions. We justify the "deformation" prescription. It is shown that the non-extremal intersecting p-brane solutions satisfy harmonic superposition rule and the intersections of non-extremal p-branes are specified by the same characteristic equations for the incidence matrices as for the extremal p-branes. We show that S-duality holds for non-extremal p-brane solutions. Generalized T-duality takes place under additional restrictions to the parameters of the theory which are the same as in the extremal case."

I.Ya.Arefeva, K.S.Viswanathan, A.I.Volovich and I.V.Volovich, "Composite p-branes in various dimensions", Nucl. Phys. Proc. Suppl. 56B (1997) 52–60

[abstract:] "We review an algebraic method of finding the composite p-brane solutions for a generic Lagrangian, in arbitrary spacetime dimension, describing an interaction of a graviton, a dilaton and one or two antisymmetric tensors. We set the Fock-De Donder harmonic gauge for the metric and the "no-force" condition for the matter fields. Then equations for the antisymmetric field are reduced to the Laplace equation and the equation of motion for the dilaton and the Einstein equations for the metric are reduced to an algebraic equation. Solutions composed of n constituent p-branes with n independent harmonic functions are given. The form of the solutions demonstrates the harmonic functions superposition rule in diverse dimensions. Relations with known solutions in D = 10 and D = 11 dimensions are discussed."

I.Ya. Aref'eva, K.S. Viswanathan and I.V. Volovich, "p-Brane solutions in diverse dimensions", Phys.Rev. D55 (1997) 4748–4755

[abstract:] "A generic Lagrangian, in arbitrary spacetime dimension, describing the interaction of a graviton, a dilaton and two antisymmetric tensors is considered. An isotropic p-brane solution consisting of three blocks and depending on four parameters in the Lagrangian and two arbitrary harmonic functions is obtained. For specific values of parameters in the Lagrangian the solution may be identified with previously known superstring solutions."

I.V. Volovich, "p-Adic string", Classical Quantum Gravity 4 (1987) 83–87

I.V. Volovich, "From p-adic strings to étale strings", Proc. Steklov Inst. Math. 203 (1995) no. 3, 37–42

I.Aref'eva and A. Volovich, "Composite p-branes in diverse dimensions", Class. Quant. Grav. 14 (1997) 2991–3000

[abstract:] "We use a simple algebraic method to find a special class of composite p-brane solutions of higher dimensional gravity coupled with matter. These solutions are composed of n constituent p-branes corresponding n independent harmonic functions. A simple algebraic criteria of existence of such solutions is presented. Relations with D = 11, 10 known solutions are discussed."

A. Volovich, "Three-block p-branes in various dimensions", Nucl. Phys. B492 (1997) 235–248

[abstract:] "It is shown that a Lagrangian, describing the interaction of the gravitation field with the dilaton and the antisymmetric tensor in arbitrary dimension spacetime, admits an isotropic p-brane solution consisting of three blocks. Relations with known p-brane solutions are discussed. In particular, in ten-dimensional spacetime the three-block p-brane solution is reduced to the known solution, which recently has been used in the D-brane derivation of the black hole entropy."

G. Dattoli, M. Del Franco, "The Euler legacy to modern physics" (preprint 09/2010)

[abstract:] "Particular families of special functions, conceived as purely mathematical devices between the end of XVIII and the beginning of XIX centuries, have played a crucial role in the development of many aspects of modern Physics. This is indeed the case of the Euler gamma function, which has been one of the key elements paving the way to string theories, furthermore the Euler–Riemann Zeta function has played a decisive role in the development of renormalization theories. The ideas of Euler and later those of Riemann, Ramanujan and of other, less popular, mathematicians have therefore provided the mathematical apparatus ideally suited to explore, and eventually solve, problems of fundamental importance in modern Physics. The mathematical foundations of the theory of renormalization trace back to the work on divergent series by Euler and by mathematicians of two centuries ago. Feynman, Dyson, Schwinger... rediscovered most of these mathematical "curiosities" and were able to develop a new and powerful way of looking at physical phenomena."

In Section 2.3 of Bernard Julia's seminal 1990 paper "Statistical theory of numbers", the author turns briefly from multiplicative to additive number theory, in particular to generating functionals associated with integer partition problems. He relates these to the Veneziano open string model, the tachyon mode, and the phenomenon of "bosonization" which is discussed elsewhere in the paper.

L. Brekke and P. Freund, "p-adic numbers in physics", Physics Reports 233, (1993) 1–66

This is a review article related to the achievements in application of p-adic numbers to string theory, quantum field theory and quantum mechanics during the period 1987-1992. The contribution of Freund and his collaborators is emphasised.

Here is an excerpt from pp.61–62:

"String theory, the candidate "theory of everything" is expected to raise fundamental issues both at the level of physics and at the level of mathematics. The old issue of the nature of continuity in physics naturally leads to the consideration of p-adic strings. It is remarkable that these very simple alternate topologies have not appeared earlier in physics (ultrametrics have appeared [62]). Yet, even now it would not be reasonable to actually select a prime and claim this to be the phenomenologically preferred prime which "underlies" physics. As we have seen, such a preferred prime could lead to serious causality problems. But if none of the primes is to be preferred, then why select a priori the prime at infinity, and deal exclusively with real numbers? A more "even-handed" procedure would envision dealing with all primes at the same time. This naturally leads to adelic theories. We have seen that this point of view immediately yields the remarkable adelic product formulae. Could it be that the adelic string is the "real thing"? This question has been raised by Manin [41] in the following (somewhat paraphrased) form. Supposing that the true physics is adelic, then why can we always assume it to be archimedean, grounded in the real numbers? Maybe this is on account of some experimental limitations, e.g. low energy. Could it be that once these limitations get lifted and we reach very high (Planck) energies, the full adelic structure of the string will reveal itself? This is an interesting possibility.

Another possibility is that the true theory is archimedean, but that on account of the product formulae, one could alternatively conceive of the theory as an Euler product over all p-adic theories. As we saw, each such theory puts the strings' world sheet on a Bethe lattice. What the adelic formulae then tell us is that we should not opt for a particular Bethe lattice as the discretization of the world sheet, but rather study absolutely all of them. The cumulative understanding of all these discretizations is tantamount to understanding the ordinary archimedean string. Of course, each of these discretizations is far simpler than the ordinary string.

On the other hand, there is the p-adics-quantum group connection, which places the ordinary and all the p-adic strings at certain special points in a continuum of theories. It is an important problem to assess the theoretical consistency of all these "quantum" strings and the phenomenological possibilities offered by them."

Note that the elements of the Monster involve Ogg's supersingular primes. Here are some instances of The Monster in string theory (thanks to Mark Thomas for pointing these out):

F. Lizzi and R.J. Szabo, "Duality symmetries and noncommutative geometry of string spacetime"

F. Lizzi and R.J. Szabo, "Noncommutative Geometry and Spacetime Gauge Symmetries of String Theory" , Chaos, Solitons and Fractals 10 (1999) 445–458

B. Craps, M.R. Gaberdiel and J.A. Harvey, "Monstrous branes", Commun. Math. Phys. 234 (2003) 229–251

M.B. Green and D. Kutasov, "Monstrous heterotic quantum mechanics", Journal of High Energy Physics 1 (01-12), 1–6

P. Henry-Labordere, B. Julia and L. Paulot, "Symmetries in M-theory: Monsters Inc." (talk given by PHL at Cargese 2002)

M.A.R. Osorio and M.A. Vazquez-Mozo, "Strings below the Planck scale", Phys. Lett. B 280 (1992) 21–25

J.A. Harvey and G. Moore, "Exact gravitational threshold correction in the FHSV model", Phys. Rev. D 57 (1998) 2329–2336

S. Chaudhuri and D.A. Lowe, "Monstrous string-string duality", Nucl. Phys. B 469 (1996) 21–36

F.D.T. Smith, "E6, strings, branes, and the standard model" (preprint 04/04)

L. Dolan and M. Langham, "Symmetric subgroups of gauged supergravities and AdS string theory vertex operators", Mod. Phys. Lett. A 14 (1999) 517–526

L. Dolan, "The beacon of Kac-Moody symmetry for physics", Notices of the American Mathematical Society, Dec. 1995, 1489

P.C. West, "Physical states and string symmetries", Mod. Phys. Lett. A 10 (1995) 761–772

K. Esmakhanova, G. Nugmanova and R. Myrzakulov, "A note on the relationship between solutions of Einstein, Ramanujan and Chazy equations" (preprint 02/2011)

[abstract:] "The Einstein equation for the Friedmann–Robertson–Walker metric plays a fundamental role in cosmology. The direct search of the exact solutions of the Einstein equation even in this simple metric case is sometime a hard job. Therefore, it is useful to construct solutions of the Einstein equation using a known solutions of some other equations which are equivalent or related to the Einstein equation. In this work, we establish the relationship the Einstein equation with two other famous equations namely the Ramanujan equation and the Chazy equation. Both these two equations play an imporatant role in the number theory. Using the known solutions of the Ramanujan and Chazy equations, we find the corresponding solutions of the Einstein equation."

[abstract:] "We obtain in this paper, as a consequence of the Riemann hypothesis, certain class of topological deformations of the graph of the function $|\zeta|$. These are used to construct an infinite set of microscopic universes (on the Planck's scale) of the Einstein type."

M. Lapidus, In Search of the Riemann Zeros (AMS, 2008)

[from publisher's description:] "In this book, the author proposes a new approach to understand and possibly solve the Riemann Hypothesis. His reformulation builds upon earlier (joint) work on complex fractal dimensions and the vibrations of fractal strings, combined with string theory and noncommutative geometry. Accordingly, it relies on the new notion of a fractal membrane or quantized fractal string, along with the modular flow on the associated modui space of fractal membranes. Conjecturally, under the action of the modular flow, the spacetime geometries become increasingly symmetric and crystal-like, hence, arithmetic. Correspondingly, the zeros of the associated zeta functions eventually condense onto the critical line, towards which they are attracted, thereby explaining why the Riemann Hypothesis must be true.

Written with a diverse audience in mind, this unique book is suitable for graduate students, experts and nonexperts alike, with an interest in number theory, analysis, dynamical systems, arithmetic, fractal or noncommutative geometry, and mathematical and mathematical or theoretical physics."

S.L. Cacciatori, M.A. Cardella, "Uniformization, unipotent flows and the Riemann hypothesis" (preprint 02/2011)

[abstract:] "We prove equidistribution of certain multidimensional unipotent flows in the moduli space of genus $g$ principally polarized abelian varieties (ppav). This is done by studying asymptotics of $\pmb{\Gamma}_{g} \sim Sp(2g,\mathbb{Z})$-automorphic forms averaged along unipotent flows, toward the codimension-one component of the boundary of the ppav moduli space. We prove a link between the error estimate and the Riemann hypothesis. Further, we prove $\pmb{\Gamma}_{g - r}$ modularity of the function obtained by iterating the unipotent average process $r$ times. This shows uniformization of modular integrals of automorphic functions via unipotent flows."

[A connection with string theory is outlined on pp.4–5.]

Conference: "Modular Forms and String Duality", Banff International Research Station, June 3–8, 2006

"Physical duality symmetries relate special limits of the various consistent string theories (Types I, II, Heterotic string and their cousins, including F-theory) one to another. By comparing the mathematical descriptions of these theories, one reveals often quite deep and unexpected mathematical conjectures. The best known string duality to mathematicians, Type IIA/IIB duality also called mirror symmetry, has inspired many new developments in algebraic and arithmetic geometry, number theory, toric geometry, Riemann surface theory, and infinite dimensional Lie algebras. Other string dualities such as Heterotic/Type II duality and F-Theory/Heterotic string duality have also, more recently, led to series of mathematical conjectures, many involving elliptic curves, K3 surfaces, and modular forms. Modular forms and quasi-modular forms play a central role in mirror symmetry, in particular, as generating functions counting the number of curves on Calabi-Yau manifolds and describing Gromov-Witten invariants. This has led to a realization that time is ripe to assess the role of number theory, in particular, that of modular forms, in mirror symmetry and string dualities in general.

One of the principal goals of this workshop is to look at modular and quasi-modular forms, congruence zeta-functions, Galois representations, and L-series for dual families of Calabi-Yau varieties with the aim of interpreting duality symmetries in terms of arithmetic invariants associated to the varieties in question. Over the last decades, a great deal of work has been done on these problems. In particular it appears that we need to modify the classical theories of Galois representations (in particular, the question of modularity) and modular forms, among others, for families of Calabi-Yau varieties in order to accommodate "quantum corrections"."

M. Nardelli, "Proposta di dimostrazione della variante Riemann di Lagarias (RH1), equivalente all'ipotesi di Riemann, con RH1=RH" (preprint in Italian, 12/2007)

[translation of abstract provided by author:] "In this paper, we suggest a proof of the Riemann Hypothesis by the 'Lagarias variant' or 'Lagarias Equivalence': RH1 = RH. Hence, we prove that for abundant and colossally abundant numbers, $L(n)$ increases progressively, as $n$ increases, although with small oscillations, but which never lead to $L(n)$ taking negative or zero values.

Furthermore, we obtain also some interesting mathematical connections between various equations concerning the Riemann zeta function and some solutions of equations regarding various models of string cosmology."

[abstract:] "This paper is a review of some interesting results that has been obtained in the study of the categories of A-branes on the dual Hitchin fibers and some interesting phenomena associated with the endoscopy in the geometric Langlands correspondence of various authoritative theoretical physicists and mathematicians."

[abstract:] "This paper is a review of some interesting results that has been obtained in the study of the quantum gravity partition functions in three-dimensions, in the Selberg zeta function, in the vanishing of cosmological constant and in the ten-dimensional anomaly cancellations. In the Section 1, we have described some equations concerning the pure three-dimensional quantum gravity with a negative cosmological constant and the pure three-dimensional supergravity partition functions. In the Section 2, we have described some equations concerning the Selberg super-trace formula for Super-Riemann surfaces, some analytic properties of Selberg super zeta-functions and multiloop contributions for the fermionic strings. In the Section 3, we have described some equations concerning the ten-dimensional anomaly cancellations and the vanishing of cosmological constant. In the Section 4, we have described some equations concerning p-adic strings, p-adic and adelic zeta functions and zeta strings. In conclusion, in the Section 5, we have described the possible and very interesting mathematical connections obtained between some equations regarding the various sections and some sectors of number t heory (Riemann zeta functions, Ramanujan modular equations, etc...) and some interesting mathematical applications concerning the Selberg super-zeta functions and some equations regarding the Section 1."

[abstract:] "This paper is a review of some interesting results that has been obtained in the study of the physical interpretation of the Riemann zeta function as a FZZT Brane Partition Function associated with a matrix/gravity correspondence and some aspects of the Rigid Surface Operators in Gauge Theory. Furthermore, we describe the mathematical connections with some sectors of String Theory (p-adic and adelic strings, p-adic cosmology) and Number Theory.

In the Section 1 we have described various mathematical aspects of the Riemann Hypothesis, matrix/gravity correspondence and master matrix for FZZT brane partition functions. In the Section 2, we have described some mathematical aspects of the rigid surface operators in gauge theory and some mathematical connections with various sectors of Number Theory, principally with the Ramanujan's modular equations (thence, prime numbers, prime natural numbers, Fibonacci's numbers, partitions of numbers, Euler's functions, etc...) and various numbers and equations related to the Lie Groups. In the Section 3, we have described some very recent mathematical results concerning the adeles and ideles groups applied to various formulae regarding the Riemann zeta function and the Selberg trace formula (connected with the Selberg zeta function), hence, we have obtained some new connections applying these results to the adelic strings and zeta strings. In the Section 4 we have described some equations concerning p-adic strings, p-adic and adelic zeta functions, zeta strings and p-adic cosmology (with regard the p-adic cosmology, some equations concerning a general class of cosmological models driven by a nonlocal scalar field inspired by string field theories). In conclusion, in the Section 5, we have showed various and interesting mathematical connections between some equations concerning the Section 1, 3 and 4."

[abstract:] "According to quantum mechanics, the properties of an atom can be calculated easily if we known the eigenfunctions and eigenvalues of quantum states in which the atom can be found. The eigenfunctions depend, in general, by the coordinates of all the electrons. However, a diagram effective and enough in many cases, we can get considering the individual eigenfunctions for individual electrons, imagining that each of them is isolated in an appropriate potential field that represent the action of the nucleus and of other electrons. From these individual eigenfunctions we can to obtain the eigenfunction of the quantum state of the atom, forming the antisymmetrical products of eigenfunctions of the individual quantum states involved in the configuration considered. The problem, with this diagram, is the calculation of the eigenfunctions and eigenvalues of individual electrons of each atomic species. To solve this problem we must find solutions to the Schroedinger's equation where explicitly there is the potential acting on the electron in question, due to the action of the nucleus and of all the other electrons of the atom. To research of potential it is possible proceed with varying degrees of approximation: a first degree is obtained by the statistical method of Thomas-Fermi in which electrons are considered as a degenerate gas in balance as a result of nuclear attraction. This method has the advantage of a great simplicity as that, through a single function numerically calculated once and for all, it is possible to represent the behaviour of all atoms. In this work (Sections 1 and 2) we give the preference to the statistical method, because in any case it provides the basis for more approximate numerical calculations. Furthermore, we describe the mathematical connections that we have obtained between certain solutions concerning the calculation of any eigenfunctions of atoms with this method, the Aurea ratio, the Fibonacci's numbers, the Ramanujan modular equations, the modes corresponding to the physical vibrations of strings, the p-adic and Adelic free relativistic particle and p-adic and adelic strings (Sections 3 and 4)."

[abstract:] "This paper is a review, a thesis, of some interesting results that has been obtained in various researches concerning the "brane collisions in string and M-theory" (Cyclic Universe), p-adic inflation and p-adic cosmology.

In Section 1 we have described some equations concerning cosmic evolution in a Cyclic Universe. In the Section 2, we have described some equations concerning the cosmological perturbations in a Big Crunch/Big Bang space-time, the M-theory model of a Big Crunch/Big Bang transition and some equations concerning the solution of a braneworld Big Crunch/Big Bang Cosmology. In the Section 3, we have described some equations concerning the generating Ekpyrotic curvature perturbations before the Big Bang, some equations concerning the effective five-dimensional theory of the strongly coupled heterotic string as a gauged version of $N = 1$ five dimensional supergravity with four-dimensional boundaries, and some equations concerning the colliding branes and the origin of the Hot Big Bang. In the Section 4, we have described some equations regarding the "null energy condition" violation concerning the inflationary models and some equations concerning the evolution to a smooth universe in an ekpyrotic contracting phase with $w > 1$. In the Section 5, we have described some equations concerning the approximateinflationary solutions rolling away from the unstable maximum of p-adic string theory. In the Section 6, we have described various equations concerning the p-adic minisuperspace model, zeta strings, zeta nonlocal scalar fields and p-adic and adelic quantum cosmology. In the Section 7, we have showed various and interesting mathematical connections between some equations concerning the p-adic Inflation, the p-adic quantum cosmology, the zeta strings and the brane collisions in string and M-theory. Furthermore, in each section, we have showed the mathematical connections with various sectors of number theory, principally the Ramanujan's modular equations, the Aurea Ratio and the Fibonacci numbers."

[abstract:] "The aim of this paper is that of show the further and possible connections between the p-adic and adelic strings and Lagrangians with Riemann zeta function with some problems, equations and theorems in number theory.

In Section 1, we have described some equations and theorems concerning the quadrature- and mean-convergence in the Lagrange interpolation. In Section 2, we have described some equations and theorems concerning the difference sets of sequences of integers. In Section 3, we have showed some equations and theorems regarding some problems of a statistical group theory (symmetric groups) and in Section 4, we have showed some equations and theorems concerning the measure of the non-monotonicity of the Euler phi function and the related Riemann zeta function.

In Section 5, we have showed some equations concerning the p-adic and adelic strings, the zeta strings and the Lagrangians for adelic strings

In conclusion, in Section 6, we have described the mathematical connections concerning the various sections previously analyzed. Indeed, in the Section 1, 2 and 3, where are described also various theorems on the prime numbers, we have obtained some mathematical connections with Ramanujan's modular equations, thence with the modes corresponding to the physical vibrations of the bosonic and supersymmetric strings and also with p-adic and adelic strings. Principally, in Section 3, where is frequently used the Hardy-Ramanujan stronger asymptotic formula and are described some theorems concerning the prime numbers. With regard Section 4, we have obtained some mathematical connections between some equations concerning the Euler phi function, the related Riemann zeta function and the zeta strings and field Lagrangians for p-adic sector of adelic string (Section 5). Furthermore, in Sections 1, 2, 3 and 4, we have described also various mathematical expressions regarding some frequency connected with the exponents of the Aurea ratio, i.e. with the exponents of the number phi = 1.61803399. We consider important remember that the number 7 of the various exponents is related to the compactified dimensions of M-theory."

[abstract:] "In this paper we have showed the various applications of the Boltzmann equation in string theory and related topics. In Section 1, we have described some equations concerning the time dependent multi-term solution of Boltzmann's equation for charged particles in gases under the influence of electric and magnetic fields, the Planck's blackbody radiation law, the Boltzmann's thermodynamic derivation and the connections with the superstring theory. In Section 2, we have described some equations concerning the modifications to the Boltzmann equation governing the cosmic evolution of relic abundances induced by dilaton dissipative-source and non-critical-string terms in dilaton-driven non-equilibrium string cosmologies. In Section 3, we have described some equations concerning the entropy of an eternal Schwarzschild black hole in the limit of infinite black hole mass, from the point of view of both canonical quantum gravity and superstring theory. We have described some equations regarding the quantum corrections to black hole entropy in string theory. Furthermore, in this section, we have described some equations concerning the thesis "Can the Universe create itself?" and the adapted Rindler vacuum in Misner space. In Section 4, we have described some equations concerning p-Adic models in Hartle-Hawking proposal and p-Adic and Adelic wave functions of the Universe. Furthermore, we have described in the various sections the various possible mathematical connections that we've obtained with some sectors of number theory and, in the Section 5, we have showed some mathematical connections between some equations of arguments above described and p-adic and adelic cosmology."

R. Turco, M. Colonnese and M. Nardelli, "On the Riemann Hypothesis. Formulas explained - $\psi(x)$ as equivalent RH. Mathematical connections with 'Aurea' section and some sectors of string theory" (preprint 06/2009)

[abstract:] "In this work we will examine the themes of RH, equivalent RH and GRH. We will explain some formulas and will show other special functions that are usually introduced with the PNT (Prime Number Theorem) and useful to investigate in other ways. In the Sections 1 and 2, we describe $\psi(x)$, i.e. the second Chebyshev function as equivalent RH. In the Section 3, we describe a step function and a generalization of Polignac. In the Section 4, we describe some equations concerning p-adic strings, p-adic and adelic zeta functions, zeta strings and zeta nonlocal scalar fields. In conclusion, in the Section 5, we have described some possible mathematical connections between adelic strings and Lagrangians with Riemann zeta function with some equations in number theory above examined."

R. Turco, M. Colonnesse, M. Nardelli, "Links between string theory and Riemann's zeta function" (preprint 01/2010)

[abstract:] "There is a connection between string theory and the Riemann's zeta function: this is an interesting way, because the zeta is related to prime numbers and we have seen on many occasions how nature likes to express himself through perfect laws or mathematical models. In [6] the authors showed all the mathematical and theoretical aspects related to the Riemann's zeta, while in [9] showed the links of certain formulas of number theory with the golden section and other areas such as string theory. The authors have proposed a solution of the Riemann hypothesis (RH) and the conjecture on the multiplicity of nontrivial zeros, showing that they are simple zeros [7][8]. Not least the situation that certain stable energy levels of atoms could be associated with non-trivial zeros of the Riemann's zeta. In [6] for example has been shown the binding of the Riemann zeta and its nontrivial zeros with quantum physics through the Law of Montgomery-Odlyzko. The law of Montgomery-Odlyzko says that "the distribution of the spacing between successive non-trivial zeros of the Riemann zeta function (normalized) is identical in terms of statistical distribution of spacing of eigenvalues in an GUE operator", which also represent dynamical systems of subatomic particles! In [10] [11] have proposed hypotheses equivalent RH, in [12] [13] the authors have presented informative articles on the physics of extra dimensions, string theory and M-theory, in [15] the conjecture Yang and Mills, in [16] the conjecture of Birch and Swinnerton-Dyer."

M. Nardelli, "From the Maxwell's equations to the string theory: new possible mathematical connections" (preprint 02/2010)

[abstract:] "In this paper in the Section 1, we describe the possible mathematics concerning the unification between the Maxwell's equations and the gravitational equations. In this Section we have described also some equations concerning the gravitomagnetic and gravitoelectric fields. In the Section 2, we have described the mathematics concerning the Maxwell's equations in higher dimension (thence Kaluza-Klein compactification and relative connections with string theory and Palumbo-Nardelli model). In the Section 3, we have described some equations concerning the noncommutativity in String Theory, principally the Dirac-Born-Infeld action, noncommutative open string actions, Chern-Simons couplings on the brane, D-brane actions and the connections with the Maxwell electrodynamics, Maxwell's equations, B-field and gauge fields. In the Section 4, we have described some equations concerning the noncommutative quantum mechanics regarding the particle in a constant field and the noncommutative classical dynamics related to quadratic Lagrangians (Hamiltonians) connected with some equations concerning the Section 3. In conclusion, in the Section 5, we have described the possible mathematical connections between various equations concerning the arguments above mentioned, some links with some aspects of Number Theory (Ramanujan modular equations connected with the physical vibrations of the superstrings, various relationships and links concerning $p, f$ thence the Aurea ratio), the zeta strings and the Palumbo-Nardelli model that link bosonic and fermionic strings".

M. Nardelli, "The mathematical theory of black holes: Mathematical connections with some sectors of string theory and number theory" (preprint 04/2010)

[abstract:] "In this paper we describe some equations concerning the stellar evolution and their stability. We describe some equations concerning the perturbations of Schwarschild blackhole, the Reissner-Nordstrom solution and the entropy of strings and black holes: Schwarzschild geometry in $D = d + 1$ dimensions. Furthermore, we show the mathematical connections with some sectors of number theory, principally with Ramanujan's modular equations and the aurea ratio (or golden ratio)."

[abstract:] "In this paper we have described, in the Section 1, some mathematics concerning the Andrica's conjecture. In the Section 2, we have described the Cramer–Shank Conjecture. In the Section 3, we have described some equations concerning the possible proof of the Cramer's conjecture and the related differences between prime numbers, principally the Cramer's conjecture and Selberg's theorem. In the Section 4, we have described some equations concerning the p-adic strings and the zeta strings. In the Section 5, we have described some equations concerning the W-deformation in toroidal compactification for N = 2 gauge theory. In conclusion, in the Section 6, we have described some possible mathematical connections between various sectors of string theory and number theory."

M. Nardelli and R. Turco, "The circle method to investigate Goldbach's conjecture and the Germain primes: Mathematical connections with p-adic strings and zeta strings" (preprint 08/2010)

[abstract:] "In this paper we have described in Section 1 some equations and theorems concerning the circle method applied to Goldbach's conjecture. In Section 2, we have described some equations and theorems concerning the circle method to investigate Germain primes by the major arcs. In Section 3, we have described some equations concerning the equivalence between Goldbach's conjecture and the generalized Riemann hypothesis. In Section 4, we have described some equations concerning p-adic strings and zeta strings. In conclusion, in Section 5, we have described some possible mathematical connections between the arguments discussed in the various sections."

[abstract:] "In this paper, in Section 1, we have described some equations concerning the functions $\zeta(s)$ and $zeta(s,w)$. In this Section, we have described also some equations concerning a transformation formula involving the gamma and Riemann zeta functions of Ramanujan. Furthermore, we have described also some mathematical connections with various theorems concerning the incomplete elliptic integrals described in "Ramanujan's lost notebook". In Section 2, we have described some Ramanujan-type series for $1/\pi$ and some equations concerning the $p$-adic open string for the scalar tachyon field. In this section, we have described also some possible and interesting mathematical connections with some Ramanujan's Theorems, contained in the first letter of Ramanujan to G.H. Hardy. In Section 3, we have described some equations concerning the zeta strings and the zeta nonlocal scalar fields. In conclusion, in Section 4, we have showed some possible mathematical connections between the arguments above mentioned, the Palumbo--Nardelli model and the Ramanujan's modular equations that are related to the physical vibrations of bosonic strings and of superstrings."

[abstract:] "In this paper, in the Section 1, we have described some equations and theorems concerning the Lebesgue integral and the Lebesgue measure. In the Section 2, we have described the possible mathematical applications, of Lebesgue integration, in some equations concerning various sectors of Chern-Simons theory and Yang-Mills gauge theory, precisely the two dimensional quantum Yang-Mills theory. In conclusion, in the Section 3, we have described also the possible mathematical connections with some sectors of String Theory and Number Theory, principally with some equations concerning the Ramanujan's modular equations that are related to the physical vibrations of the bosonic strings and of the superstrings, some Ramanujan's identities concerning $\pi$ and the zeta strings."

[abstract:] "This paper is principally a review, a thesis, of principal results obtained from various authoritative theoretical physicists and mathematicians in some sectors of theoretical physics and mathematics. In this paper in the Section 1, we have described some equations concerning the quantum electrodynamics coupled to quantum gravity. In the Section 2, we have described some equations concerning the gravitational contributions to the running of gauge couplings. In the Section 3, we have described some equations concerning some quantum effects in the theory of gravitation. In the Section 4, we have described some equations concerning the supersymmetric Yang-Mills theory applied in string theory and some lemmas and equations concerning various gauge fields in any non-trivial quantum field theory for the pure Yang-Mills Lagrangian. Furthermore, in conclusion, in the Section 5, we have described various possible mathematical connections between the argument above mentioned and some sectors of Number Theory and String Theory, principally with some equations concerning the Ramanujan's modular equations that are related to the physical vibrations of the bosonic strings and of the superstrings, some Ramanujan's identities concerning $\pi$ and the zeta strings."

[abstract:] "In this paper in the Section 1, we have described some equations concerning the duality and higher derivative terms in M-theory. In the Section 2, we have described some equations concerning the moduli-dependent coefficients of higher derivative interactions that appear in the low energy expansion of the four-supergraviton amplitude of maximally supersymmetric string theory compactified on a d-torus. Thence, some equations regarding the automorphic properties of low energy string amplitudes in various dimensions. In the Section 3, we have described some equations concerning the Eisenstein series for higher-rank groups, string theory amplitudes and string perturbation theory. In the Section 4, we have described some equations concerning U-duality invariant modular form for the D^6R^4 interaction in the effective action of type IIB string theory compactified on T^2. Furthermore, in the Section 5, we have described various possible mathematical connections between the arguments above mentioned and some sectors of Number Theory, principally the Aurea Ratio Phi, some equations concerning the Ramanujan's modular equations that are related to the physical vibrations of the bosonic strings and of the superstrings, some Ramanujan's identities concerning p and the zeta strings. In conclusion, in the Appendix A, we have analyzed some pure numbers concerning various equations described in the present paper. Thence, we have obtained some useful mathematical connections with some sectors of Number Theory. In the Appendix B, we have showed the column "system" concerning the universal music system based on Phi and the table where we have showed the difference between the values of Phi^(n/7) and the values of the column "system"."

[abstract:] "The present paper is a review, a thesis of some very important contributes of E. Witten, C. Beasley, R. Ricci, B. Basso et al. regarding various applications concerning the Jones polynomials, the Wilson loops and the cusp anomaly and integrability from string theory. In this work, in Section 1, we have described some equations concerning the knot polynomials, the Chern–Simons from four dimensions, the D3-NS5 system with a theta-angle, the Wick rotation, the comparison to topological field theory, the Wilson loops, the localization and the boundary formula. We have described also some equations concerning electric-magnetic duality to $N = 4$ super Yang-Mills theory, the gravitational coupling and the framing anomaly for knots. Furthermore, we have described some equations concerning the gauge theory description, relation to Morse theory and the action. In Section 2, we have described some equations concerning the applications of non-abelian localization to analyze the Chern–Simons path integral including Wilson loop insertions. In the Section 3, we have described some equations concerning the cusp anomaly and integrability from string theory and some equations concerning the cusp anomalous dimension in the transition regime from strong to weak coupling. In Section 4, we have described also some equations concerning the "fractal" behaviour of the partition function.

Also here, we have described some mathematical connections between various equation described in the paper and (i) the Ramanujan's modular equations regarding the physical vibrations of the bosonic strings and the superstrings, thence the relationship with the Palumbo-Nardelli model, (ii) the mathematical connections with the Ramanujan's equations concerning $\pi$ and, in conclusion, (iii) the mathematical connections with the golden ratio $\phi$ and with $1.375$ that is the mean real value for the number of partitions $p(n)$."

[abstract:] "The present paper is a review, a thesis of some very important contributes of P. Horava, M. Fabinger, M. Bordag, U. Mohideen, V.M. Mostepanenko, Trang T. Nguyen et al. regarding various applications concerning the Casimir Effect.

In this paper in the Section 1 we have showed some equations concerning the Casimir Effect between two ends of the world in M-theory, the Casimir force between the boundaries, the Casimir effect on the open membrane, the Casimir form and the Casimir correction to the string tension that is finite and negative. In the Section 2, we have described some equations concerning the Casimir effect in spaces with nontrivial topology, i.e. in spaces with non-Euclidean topology, the Casimir energy density of a scalar field in a closed Friedmann model, the Casimir energy density of a massless field, the Casimir contribution and the total vacuum energy density, the Casimir energy density of a massless spinor field and the Casimir stress-energy tensor in the multi-dimensional Einstein equations with regard the Kaluza–Klein compactification of extra dimensions.

Further, in the Section 1 and 2 we have described some mathematical connections concerning some sectors of Number Theory, i.e. the Palumbo-Nardelli model, the Ramanujan modular equations concerning the physical vibrations of the bosonic strings and the superstrings and the connections of some values contained in the equations with some values concerning the new universal music system based on fractional powers of Phi and Pigreco.

In the Section 3, we have described some mathematical connections concerning the Riemann zeta function and the zeta-strings. In conclusion, in Section 4, we have described some mathematical connections concerning some equations regarding the Casimir effect and vacuum fluctuations. In conclusion (Appendix A), we have described some mathematical connections between the equation of the energy negative of the Casimir effect, the Casimir operators and some sectors of number theory, i.e. the triangular numbers, the Fibonacci numbers, phi, Pigreco and the partition of numbers."

[abstract:] "In the present paper we have described some interesting mathematical applications of number theory to heterotic string theory $E8 \times E8$. In Chapter 1, we have described various theoretic arguments and equations concerning the Lie group $E8$, $E8 \times E8$ gauge fields and heterotic string theory. In Chapter 2, we have described the link between the subsets of odd natural numbers and of squares, some equations concerning the theorem that: 'every sufficiently large odd positive integer can be written as the sum of three primes', and the possible method of factorization of a number. In Chapter 3, we have described some classifications of the numbers: perfect, defective, abundant. Furthermore, we have described an infinite set of integers, each of which has many factorizations. In Chapter 4, we have described some interesting mathematical applications concerning the possible method of factorization of a number to the number of dimensions of the Lie group $E8$. In conclusion, in the Appendix, we have described some mathematical connections between various series of numbers concerning Chapter 1 and some sectors of number theory."

P.F. Roggero, M. Nardelli and F. Di Noto, "Study on the Riemann zeta function" (preprint 11/2012)

[abstract:] "In this paper we show some connections between hyperbolic cotangent and Riemann zeta function plus many other interesting relations. Furthermore, we show also some possible mathematical connections between some equations concerning this thesis and some equations regarding the zeta-strings and the zeta nonlocal scalar fields."

[abstract:] "In this paper we show that perfect numbers are only 'even' plus many other interesting relations about Mersenne's prime. Furthermore, we describe also various equations, lemmas and theorems concerning the expression of a number as a sum of primes and the primitive divisors of Mersenne numbers. In conclusion, we show some possible mathematical connections between some equations regarding the arguments above mentioned and some sectors of string theory ($p$-adic and adelic strings and Ramanujan modular equation linked to the modes corresponding to the physical vibrations of the bosonic strings)."

M. Nardelli, "A possible proof of Fermat's Last Theorem throught the abc radical" (preprint 03/2013)

[abstract:] "In this paper we show a possible proof of Fermat's Last Theorem through the 'abc' radical. Furthermore, in the various sections, we have described also some mathematical connections with $\pi$, $\phi$, thence with some sectors of string theory."

[abstract:] "In the present paper in the Section 1, we have described some equations concerning the cusp anomalous dimension in the planar limit of $N = 4$ super Yang–Mills from a thermodynamic Bethe ansatz (TBA) system, the Luscher correction at strong coupling and the strong coupling expansion of the function $F$. In Section 2, we have described some equations concerning a two-parameter family of Wilson loop operators in $N = 4$ supersymmetric Yang–Mills theory which interpolates smoothly between the $1/2$ BPS line or circle, principally some equations concerning the one-loop determinants. In Section 3, we have described some results and equations of the mathematician Ramanujan concerning some definite integrals and an infinite product and some equations concerning the development of derivatives of order $n$ ($n$ positive integer) of various trigonometric functions and divergent series. Thence, we have described some mathematical connections between some equations concerning this section and Sections 1 and 2. In Section 4, we have described some equations concerning the relationship between Yang–Mills theory and gravity and, consequently, the complete four-loop four-point amplitude of $N = 4$ super-Yang–Mills theory including the nonplanar contributions regarding the gauge theory and the gravity amplitudes. In conclusion, in the Appendix A and B, we have described a new possible method of factorization of a number and various mathematical connections with some sectors of Number Theory (Fibonacci numbers, Lie numbers, triangular numbers, $\Phi$, $\pi$, etc.)."

P.F. Roggero, M. Nardelli and F. Di Noto, "Universal rule to find all the prime numbers" (preprint 09/2013)

[abstract:] "In the present paper in Section 1, we have described the formula to find all the prime numbers. In Section 1.1, we have described some equations and lemmas concerning the prime numbers and various mathematical connections with some sectors of string theory. In Section 2, we have described the universal rule to find a prime number as large as desired."

P.F. Roggero, M. Nardelli and F. Di Noto, "Relations between the Gauss–Eisenstein prime numbers and their correlation with Sophie Germain primes" (preprint 11/2013)

[abstract:] "In the present paper we examine the relations between the Gauss prime numbers and the Eisenstein prime numbers and their correlation with Sophie Germain primes. Furthermore, we have described also various mathematical connections with some equations concerning the string theory."

O. Volonterio, M. Nardelli and F. Di Noto, "On a new mathematical application concerning the discrete and the analytic functions. Mathematical connections with some sectors of number theory and string theory" (preprint 02/2014)

[abstract:] "In this work we have described a new mathematical application concerning discrete and analytic functions: the Volonterio transform and the Volonterio polynomial. The Volonterio transform (V transform), indeed, works from the world of discrete functions to the world of analytic functions. We have described various mathematical applications and properties of them. Furthermore, we have showed also various examples and the possible mathematical connections with some sectors of number theory and string theory."

P. F. Roggero, M. Nardelli and F. Di Noto, "On some equations concerning Riemann's prime number formula and on a secure and efficient primality test. Mathematical connections with some sectors of string theory" (preprint 06/2014)

[abstract:] "In this paper we focus attention on some equations concerning Riemann's prime number formula and on the behavior of a secure primality test. Furthermore, we have described also some mathematical connections with some sectors of string theory."

O. Volonterio and M. Nardelli, "On some applications of the Volonterio transform: Series development of type $Nk+M$ adn mathematical connections with some sectors of string theory" (preprint 02/2015)

[abstract:] "In this work we have described a new mathematical application concerning discrete and the analytic functions: the Volonterio transform (V transform) and the Volonterio polynomial. We have descrive various mathematical applications and properties of them, precisely the series development of the type $Nk+M$. Furthermore, we have showed also various examples and the possible mathematical connections with some sectors of number theory and string theory."

M. Nardelli and R. Servi, "On some equations concerning the M-theory and topological strings and the Gopakumar–Vafa formula applied in some sectors of string theory and number theory" (preprint 05/2015)

[abstract:] "In the present paper we have described in Chapter 1 some equations concerning M-Theory, topological strings and topological gauge theory, in Chapter 2 some equations concerning the Gopakumar–Vafa formula in Type IIA compactification to four dimensions on a Calabi–Yau manifold in terms of a counting of BPS states in M-theory. Finally, in Chapter 3, we have described some possible methods of factorization and their various possible mathematical connections concerning the solutions for some equations regarding the above sectors of string theory."

M. Nardelli, F. Di Noto and P. Roggero, "On some mathematical connections between the cubic equation and some sectors of string theory and relativistic quantum gravity" (preprint 11/2015)

[abstract:] "In this paper we have described some interesting mathematical connections with various expressions of some sectors of string theory and relativistic quantum gravity, principally with the Palumbo–Nardelli model applied to the bosonic strings and the superstrings, and some parts of the theory of the cubic equation. In Appendix A, we have described the mathematical connections with some equations concerning the possible relativistic theory of quantum gravity. In conclusion in Appendix B, we have described a proof of Fermat's Last Theorem for the cubic equation case $n = 3$."

P. Roggero, M. Nardelli and F. Di Noto, "The sum of reciprocal Fibonacci prime numbers converges to a new constant: Mathematical connections with some sectors of Einstein's field equations and string theory" (preprint 03/2016)

[abstract:] "In this paper we have described a sum of the reciprocal Fibonacci primes that converges to a new constant. Furthermore, in the Section 2, we have described also some new possible mathematical connections with the universal gravitational constant $G$, the Einstein field equations and some equations of string theory linked to $\phi$ and $\pi$."

P. Roggero, M. Nardelli and F. Di Noto, "Sum of the reciprocals of famous series: Mathematical connections with some sectors of theoretical physics and string theory" (preprint 01/2017)

[abstract:] "In this paper it has been calculated the sums of the reciprocals of famous series. The sum of the reciprocals gives fundamental information on these series. The higher this sum and larger numbers there are in series and vice versa. Furthermore we understand also what is the growth factor of the series and that there is a clear link between the sums of the reciprocal and the "intrinsic nature" of the series. We have described also some mathematical connections with some sectors of theoretical physics and string theory."

Nardelli has also provided four other preprints involving work relating aspects of number theory to string theory, quantum cosmology, gauge theory, noncommutative geometry, etc.: [1]   [2]   [3]   [4]

Here is an excerpt from a posting by on the sci.physics newsgroup (02/98) by Dan Piponi:

"In (bosonic) string theory via the operator formalism you find an infinite looking zero point energy just like in QED except that you get a sum that looks like:

1+2+3+4+...

Now the naive thing to do is the same: subtract off this zero point energy. However later on you get into complications. In fact (if I remember correctly) you must replace this infinity with -1/12 (of all things!) to keep things consistent.

Now it turns out there is a nice mathematical kludge that allows you to see 1+2+3+4+... as equalling -1/12. What you do is rewrite it as

1+2-n +3-n +...

This is the Riemann Zeta function. This converges for large n but can be analytically continued to n = -1, even though the series doesn't converge there. Zeta(-1) is -1/12. So in some bizarre sense 1+2+3+4+... really is -1/12.

But even more amazingly is that you can get the -1/12 by a completely different route - using the path integral formalism rather than the operator formalism. This -1/12 is tied up in a deep way with the geometry of string theory so it's a lot more than simply a trick to keep the numbers finite.

However I don't know if the equivalent operation in QED is tied up with the same kind of interesting geometry."

C. Clark, "Math formula gives new glimpse into the magical mind of Ramanujan" (phys.org, 12/2012)

[excerpt:] "The result is a formula for mock modular forms that may prove useful to physicists who study black holes. The work, which Ono recently presented at the Ramanujan 125 conference at the University of Florida, also solves one of the greatest puzzles left behind by the enigmatic Indian genius.

"No one was talking about black holes back in the 1920s when Ramanujan first came up with mock modular forms, and yet, his work may unlock secrets about them," [Ken] Ono says. Expansion of modular forms is one of the fundamental tools for computing the entropy of a modular black hole. Some black holes, however, are not modular, but the new formula based on Ramanujan's vision may allow physicists to compute their entropy as though they were."

J.F.R. Duncan, M.J. Griffin and K. Ono, "Moonshine" (preprint 11/2014)

[abstract:] "Monstrous moonshine relates distinguished modular functions to the representation theory of the monster. The celebrated observations that 196884 = 1 + 196883 and 21493760 = 1+196883+21296876, etc., illustrate the case of the modular function j - 744, whose coefficients turn out to be sums of the dimensions of the 194 irreducible representations of the monster. Such formulas are dictated by the structure of the graded monstrous moonshine modules. Recent works in moonshine suggest deep relations between number theory and physics. Number theoretic Kloosterman sums have reappeared in quantum gravity, and mock modular forms have emerged as candidates for the computation of black hole degeneracies. This paper is a survey of past and present research on moonshine. We also obtain exact formulas for the multiplicities of the irreducible components of the moonshine modules. These formulas imply that such multiplicities are asymptotically proportional to dimensions."

number theory, renormalisation and zeta-function regularisation techniques

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