(preprint 06/2019)
[abstract:] "In this paper, we proposed a novel method using the elementary number theory to investigate the discrete nature of the screw dislocations in crystal lattices, simple cubic (SC) lattice and body centered cubic (BCC) lattice, by developing the algebraic description of the dislocations in the previous report (Hamada, Matsutani, Nakagawa, Saeki, Uesaka, Pacific J. Math. for Industry 10 (2018), 3). Using the method, we showed that the stress energy of the screw dislocations in the BCC lattice and the SC lattice are naturally described; the energy of the BCC lattice was expressed by the truncated Epstein–Hurwitz zeta function of the Eisenstein integers, whereas that of SC lattice is associated with the truncated Epstein–Hurwitz zeta function of the Gauss integers."
M. Nixon, M. Fridman, E. Ronen, A.A. Friesem, N. Davidson and I. Kanter, "Controlling synchronization in large laser networks using number theory" (preprint 12/2011)
[abstract:] "Synchronization in networks with delayed coupling are ubiquitous in nature and play a key role in almost all fields of science including physics, biology, ecology, climatology and sociology. In general, the published works on network synchronization are based on data analysis and simulations, with little experimental verification. Here we develop and experimentally demonstrate various multi-cluster phase synchronization scenarios within coupled laser networks. Synchronization is controlled by the network connectivity in accordance to number theory, whereby the number of synchronized clusters equals the greatest common divisor of network loops. This dependence enables remote switching mechanisms to control the optical phase coherence among distant lasers by local network connectivity adjustments. Our results serve as a benchmark for a broad range of coupled oscillators in science and technology, and offer feasible routes to achieve multi-user secure protocols in communication networks and parallel distribution of versatile complex combinatorial tasks in optical computers."
L.F. Alday and J.-B. Bae, "Rademacher expansions and the spectrum of 2d CFT" (preprint 12/2019)
[abstract:] "A classical result from analytic number theory by Rademacher gives an exact formula for the Fourier coefficients of modular forms of non-positive weight. We apply similar techniques to study the spectrum of two-dimensional unitary conformal field theories, with no extended chiral algebra and $c > 1$. By exploiting the full modular constraints of the partition function we propose an expression for the spectral density in terms of the light spectrum of the theory. The expression is given in terms of a Rademacher expansion, which converges for spin $j \neq 0$. For a finite number of light operators the expression agrees with a variant of the Poincare construction developed by Maloney, Witten and Keller. With this framework we study the presence of negative density of states in the partition function dual to pure gravity, and propose a scenario to cure this negativity."
L. Dal Negro, Y. Chen and F. Sgrignuoli, "Aperiodic photonics of elliptic curves" (preprint 12/2019)
[abstract:] "In this paper we propose a novel approach to aperiodic order in optical science and technology that leverages the intrinsic structural complexity of certain non-polynomial (hard) problems in number theory and cryptography for the engineering of optical media with novel transport and wave localization properties. In particular, we address structure-property relationships in a large number (900) of light scattering systems that physically manifest the distinctive aperiodic order of elliptic curves and the associated discrete logarithm problem over finite fields. Besides defining an extremely rich subject with profound connections to diverse mathematical areas, elliptic curves offer unprecedented opportunities to engineer light scattering phenomena in aperiodic environments beyond the limitations of traditional random media. Our theoretical analysis combines the interdisciplinary methods of point patterns spatial statistics with the rigorous Green's matrix solution of the multiple wave scattering problem for electric and magnetic dipoles and provides access to the spectral and light scattering properties of novel deterministic aperiodic structures with enhanced light-matter coupling for nanophotonics and metamaterials applications to imaging and spectroscopy."
Y. Nellambakam and K.V.S. Shiv Chaitanya, "Negative refractive index, perfect lens and Cesàro convergence" (preprint 05/2020)
[abstract:] "In this letter, we show that the restoration of evanescent wave in perfect lens obeys a new kind of convergence known as Cesaro convergence. Cesaro convergence allows us to extend the domain of convergence that is analytically continuing to the complex plane in terms of Riemann zeta function. Therefore, from the properties of Riemann zeta function we show that it is not possible to restore the evanescent wave for all the values of $r'_z$, [here $r'_z$ is complex]. The special value, that is, $r'_z = 1 = 2+ib$ refers to the non-existence of evanescent wave, is the physicists proof of Riemann Hypothesis."
Y. Nellambakam and K.V.S. Shiv Chaitanya, "Metamaterials and Cesàro convergence" (preprint 01/2020)
[abstract:] "In this paper, we show that the linear dielectrics and magnetic materials in matter obey a special kind of mathematical property known as Cesàro convergence. Then, we also show that the analytical continuation of the linear permittivity and permeability to a complex plane in terms of Riemann zeta function. The metamaterials are fabricated materials with a negative refractive index. These materials, in turn, depend on permittivity and permeability of the linear dielectrics and magnetic materials. Therefore, the Cesàro convergence property of the linear dielectrics and magnetic materials may be used to fabricate the metamaterials."
I.D. Mayergoyz, "Plasmon Resonances in Nanoparticles, Their Applications to Magnetics and Relation to the Riemann Hypothesis" (preprint 04/2011)
[abstract:] "The review of the mathematical treatment of plasmon resonances as an eigenvalue problem for specific boundary integral equations is presented and general properties of plasmon spectrum are outlined. Promising applications of plasmon resonances to magnetics are described. Interesting relation of eigenvalue treatment of plasmon resonances to the Riemann hypothesis is established."
S. Boatto and J. Koiller, "Vortices on closed surfaces" (preprint 02/2008)
[abstract:] "We consider $N$ point vortices $s_j$ of strengths $\kappa_j$ moving on a closed (compact, boundaryless, orientable) surface $S$ with riemannian metric $g$. As far as we know, only the sphere or surfaces of revolution, the latter qualitatively, have been treated in the available literature. The aim of this note is to present an intrinsic geometric formulation for the general case. We give a simple proof of Kimura's conjecture that a dipole describes geodesic motion. Searching for integrable vortex pairs systems on Liouville surfaces is in order. The vortex pair system on a triaxial ellipsoid extends Jacobi's geodesics. Is it Arnold–Liouville integrable? Not in our wildest dreams is another possibility: that quantizing a vortex system could relate with a million dollars worth question, but we took courage – nerve is more like it – to also present it."
[The paper concludes with the following suggestion that there might be a connection to the Riemann Hypothesis:
"A connection may exist between the zeros of Riemann's zeta function and the quantization of a 3/2
degrees of freedom vortex monopole problem on some compact Riemann surface..."]
Y. Hashimoto, "Correlations of multiplicities in length spectra for congruence subgroups" (preprint 02/2012)
[abstract:] "Bogomolny-Leyvraz-Schmit (1996) and Peter (2002) proposed an asymptotic formula for the correlation of the multiplicities in length spectrum on the modular surface, and Lukianov (2007) extended its asymptotic formula to the Riemann surfaces derived from the congruence subgroup $\Gam_0(n)$ and the quaternion type co-compact arithmetic groups. The coefficients of the leading terms in these asymptotic formulas are described in terms of Euler products over prime numbers, and they appear in eigenvalue statistic formulas found by Rudnick (2005) and Lukianov (2007) for the Laplace-Beltrami operators on the corresponding Riemann surfaces. In the present paper, we further extend their asymptotic formulas to the higher level correlations of the multiplicities for any congruence subgroup of the modular group."
R. Movassagh, Y. Tsuji and R. Hoffmann, "The exact form of the Green's function of the Hückel (tight binding) model" (preprint 07/2014)
[abstract:] "The applications of the Hückel (tight binding) model are ubiquitous in quantum chemistry and solid state physics. The matrix representation is isomorphic to an unoriented vertex adjacency matrix of a bipartite graph, which is also the Laplacian matrix plus twice the identity. In this paper, we analytically calculate the determinant and, when it exists, the inverse of this matrix in connection with the Green's function, $G$, of the $N \times N$ Hückel matrix for linear chains and cyclic systems. For an open linear chain we prove that $G$ is a real symmetric matrix whose entries are $G(r,s) = (-1)^{\frac{r+s-1}{2}}$ when $r$ is even and $s < r$ is odd; $G(r,s)=0$ otherwise. A corollary is a closed form expression for a Harmonic sum. For a ring we calculate the inverse, give formulas for the entries and find that it is always a Toeplitz matrix. We then extend the results to $d$-dimensional lattices, whose linear size is $N$. The existence of the inverse becomes a question of number theory. We prove that the inverse exists if and only if $N+1$ and $d$ are odd and $d$ is smaller than the smallest divisor of $N+1$. We corroborate our results by numerical demonstrations of the entry patterns of the Green's function and discuss applications related to transport and conductivity."
"A Trillion Triangles", AIM News (22/09/09)
"In addition to the practical advances required for this result, the answer also has theoretical implications. According to mathematician Michael Rubinstein from the University of Waterloo, "A few years ago we combined ideas from number theory and physics to predict how congruent numbers behave statistically. I was very pleased to see that our prediction was quite accurate.""
K. Nakayama, F. Takahashi and T.T. Yanagida, "Number theory dark matter" (preprint 02/2011)
[abstract:] "We propose that the stability of dark matter is ensured by a discrete subgroup of the U(1)B-L gauge symmetry, Z_2(B-L). We introduce a set of chiral fermions charged under the U(1)B-L in addition to the right-handed neutrinos, and require the anomaly-cancellation conditions associated with the U(1)B-L gauge symmetry. We find that the possible number of fermions and their charges are tightly constrained, and that non-trivial solutions appear when at least five additional chiral fermions are introduced. The Fermat theorem in the number theory plays an important role in this argument. Focusing on one of the solutions, we show that there is indeed a good candidate for dark matter, whose stability is guaranteed by Z_2(B-L)."
Bo Hu, "Neutrino mixing and discrete symmetries" (preprint 12/2012)
[abstract:] "A model independent study of neutrino mixing based on a new method to derive mixing patterns is presented. An interesting result we find is that, in the case where unbroken residual symmetries of the Majorana neutrino and left-handed charged-lepton mass matrices obey some general assumptions, the complete set of possible mixing patterns can be determined by the solutions to the constraint equation with the help of algebraic number theory. This method can also be applied to more general cases beyond the minimal scenario. Several applications and phenomenological implications are discussed."
C. King, "Experimental observations on the Riemann hypothesis, and the Collatz conjecture" (preprint 05/2010)
[abstract:] "This paper seeks to explore whether the Riemann hypothesis falls into a class of putatively unprovable mathematical conjectures, which arise as a result of unpredictable irregularity. It also seeks to provide an experimental basis to discover some of the mathematical enigmas surrounding these conjectures, by providing Matlab and C programs which the reader can use to explore and better understand these systems."
C. Moore and A. Russell, "Approximate representations and approximate homomorphisms" (preprint 09/2010)
[abstract:] "Approximate algebraic structures play a defining role in arithmetic combinatorics and have found remarkable applications to basic questions in number theory and pseudorandomness. Here we study approximate representations of finite groups: functions f:G -> U_d such that Pr[f(xy) = f(x) f(y)] is large, or more generally Exp_{x,y} ||f(xy) - f(x)f(y)||^2 is small, where x and y are uniformly random elements of the group G and U_d denotes the unitary group of degree d. We bound these quantities in terms of the ratio d / d_min where d_min is the dimension of the smallest nontrivial representation of G. As an application, we bound the extent to which a function f : G -> H can be an approximate homomorphism where H is another finite group. We show that if H's representations are significantly smaller than G's, no such f can be much more homomorphic than a random function.
We interpret these results as showing that if G is quasirandom, that is, if d_min is large, then G cannot be embedded in a small number of dimensions, or in a less-quasirandom group, without significant distortion of G's multiplicative structure. We also prove that our bounds are tight by showing that minors of genuine representations and their polar decompositions are essentially optimal approximate representations."
B. Luque, O. Miramontes and L. Lacasa, "Number theoretic example of scale-free topology inducing self-organized criticality", Phys. Rev. Lett. 101 (2008) 158702
[abstract:] "In this Letter we present a general mechanism by which simple dynamics running on networks become self-organized critical for scale-free topologies. We illustrate this mechanism with a simple arithmetic model of division between integers, the division model. This is the simplest self-organized critical model advanced so far, and in this sense it may help to elucidate the mechanism of self-organization to criticality. Its simplicity allows analytical tractability, characterizing several scaling relations. Furthermore, its mathematical nature brings about interesting connections between statistical physics and number theoretical concepts. We show how this model can be understood as a self-organized stochastic process embedded on a network, where the onset of criticality is induced by the topology."
D. Vilone, "Complex network approach to number theory" (preprint 09/2014)
[abstract:] "In this short paper, following the most recent advances in complex network theory, a new approach to number theory with potential applications to other fields is proposed. The model by Garcia-Perez, Serrano and Boguna, introduces an algorithm which allows to create a bipartite graph of integers (with primes and composites) statistically very close to the real one. Since the algorithm is defined a priori, we can have a description of the simulated prime number distribution in terms of a known differential equation, which in general can be treated more easily. The so determined properties of the simulated distribution can give useful hints about the behavior of the real prime number distribution. In principle it could be also possible to demonstrate open questions in number theory, proven the total equivalence of the simulated and real distributions. I show that the model by Garcia-Perez, Serrano and Boguna, though very good to describe the architecture of the relations among composites and primes, can not be useful to catch the most subtle properties of prime numbers. Anyway, the path suggested by such model is very promising, and should drive researchers to look for more refined algorithms to simulate the structure of integer numbers."
V.L.Cartas, "The Riemann zeta function
applied to the glassy systems and neural networks" (presented at International Conference on Theoretical Physics, Paris, 22–27 July 2002)
[Abstract:] "In the present paper it is described how the Riemann zeta function could be a very useful tool
in the analyze of the glassy systems and the neural networks. According to A. Crisanti and F. Ritort, this kind of
complex systems could be analyzed using a simple solvable model of glass: "The oscillator model" which is
defined by a set of N non-interacting harmonic oscillators with energy. The Riemann zeta function is
used to describe the Crisanti-Ritort System. It has been also made a topological study in order to have a more
intuitive representation of the critical points, where the states of the system changes."
P. Le Doussal, Z. Ristivojevic and K.J. Wiese, "Exact form of the exponential correlation function in the glassy super-rough phase" (preprint 04/2013)
[abstract:] "We consider the random-phase sine-Gordon model in two dimensions. It describes two-dimensional elastic systems with random periodic disorder, such as pinned flux-line arrays, random field XY models, and surfaces of disordered crystals. The model exhibits a super-rough glass phase at low temperature $T < T_{c}$ with relative displacements growing with distance $r$ as $\bar{\langle [\theta(r)-\theta(0)]^2\rangle} \simeq A(\tau) \ln^2 (r/a)$, where $A(\tau) = 2 \tau^2- 2 \tau^3 +\mathcal{O}(\tau^4)$ near the transition and $\tau=1-T/T_{c}$. We calculate all higher cumulants and show that they grow as $\bar{\langle[\theta(r)-\theta(0)]^{2n}\rangle}_c \simeq [2 (-1)^{n+1} (2n)! \zeta(2n-1) \tau^2 + \mathcal{O}(\tau^3) ] \ln(r/a)$, $n \geq 2$, where $\zeta$ is the Riemann zeta function. By summation we obtain the decay of the exponential correlation function as $\bar{\langle e^{iq\left[\theta(r)-\theta(0)\right]}\rangle} \simeq (a/r)^{\eta(q)} \exp(-\frac{1}{2}\mathcal{A}(q)\ln^2 (r/a))$ where $\eta(q)$ and ${\cal A}(q)$ are obtained for arbitrary $q \leq 1$ to leading order in $\tau$. The anomalous exponent is $\eta(q) = c q^2 - \tau^2 q^2 [2\gamma_E+\psi(q)+\psi(-q)]$ in terms of the digamma function $\psi$, where $c$ is non-universal and $\gamma_E$ is the Euler constant. The correlation function shows a resonant deep at $q=1$, corresponding to fermion operators in the dual picture, which should bevisible in Bragg scattering experiments."
D. Merlini, "The Riemann magneton of
the primes" (preprint 04/04)
[abstract:] "We present a calculation involving a function related
to the Riemann Zeta function and suggested by two recent works
concerning the Riemann Hypothesis: one by Balazard, Saias and Yor and
the other by Volchkov. We define an integral m(r)
involving the Zeta function in the complex variable s = r
+ it and find a particularly interesting expression for
m(r) which is rigorous at least in some range of r.
In such a range we find that there are two discontinuities of the
derivative m'(r) at r = 1 and r = 0, which
we calculate exactly. The jump at r = 1 is given by 4*Pi. The
validity of the expression for m(r) up to r =
1/2 is equivalent to the truth of the Riemann Hypothesis (RH).
Assuming RH the expression for m (r) gives m = 0
at r = 1/2 and the slope m'(r) = Pi*(1 + gamma) =
4.95 at r = 1/2 (where gamma = 0.577215... is the Euler
constant). As a consequence, if the expression for m(r)
can be continued up to r = 1/2, then if we interpret
m(r) as a magnetization in the presence of a magnetic
field h = r - 1/2 (or as a "free energy" at inverse
temperature beta proportional to r), there is a first order
phase transition at r = 1/2 (h = 0) with a jump of
m'(r) given by 2*Pi times the first Lin coefficient
lambda_1 = [1+gamma/2-(1/2)ln(4*Pi)] = 0.0230957. Independently of the
RH, by looking at the behavior of the convergent Taylor expansion of
m(r) at r = 1-, m(r = 1/2+) as well
as the first Lin coefficient may be evaluated using the Euler product
formula, in terms of the primes. This gives further evidence for the
possible truth of the Riemann Hypothesis."
S. Beltraminelli, D. Merlini, S. Sekatskii, "A hidden symmetry
related to the Riemann Hypothesis with the primes into the critical Strip" (preprint 03/2008)
[abstract:] "In this note concerning integrals involving the logarithm of the Riemann Zeta function, we extend some treatments
given in previous pioneering works on the subject and introduce a more general set of Lorentz measures. We first obtain two new
equivalent formulations of the Riemann Hypothesis (RH). Then with a special choice of the measure we formulate the RH as a
'hidden symmetry', a global symmetry which connects the region outside the critical strip with that inside the critical strip. The
Zeta function with all the primes appears as argument of the Zeta function in the critical strip. We then illustrate the treatment by a
simple numerical experiment. The representation we obtain go a little more in the direction to believe that RH may eventually be true."
H. Kösters, "The Riemann zeta-function and the sine kernel"
(preprint 03/2008)
[abstract:] "We point out an interesting occurrence of the sine kernel in connection with the shifted moments of the Riemann
zeta-function along the critical line. We establish this occurrence rigorously for the shifted second moment and, under some constraints
on the shifts, for the shifted fourth moment. Our proofs of these results closely follow the classical proofs for the non-shifted moments of
the Riemann zeta-function. Furthermore, we conjecture that the sine kernel also occurs in connection with the higher (even) shifted
moments and show that this conjecture is closely related to a recent conjecture by Conrey, Farmer, Keating, Rubinstein, and Snaith."
G.A. Hiary and M.O. Rubinstein, "Uniform asymptotics for the full moment conjecture of the Riemann zeta function" (preprint 06/2011)
[abstract:] "Conrey, Farmer, Keating, Rubinstein, and Snaith recently conjectured formulas for the full asymptotics of the moments of $L$-functions. In the case of the Riemann zeta function, their conjecture states that the $2k$-th absolute moment of zeta on the critical line is asymptotically given by a certain $2k$-fold residue integral. This residue integral can be expressed as a polynomial of degree $k^2$, whose coefficients are given in exact form by elaborate and complicated formulas. In this article, uniform asymptotics for roughly the first $k$ coefficients of the moment polynomial are derived. Numerical data to support our asymptotic formula are presented. An application to bounding the maximal size of the zeta function is considered."
E.C. Bailey and J.P. Keating, "On the moments of the moments of $\zeta(1/2+it)$" (preprint 06/2020)
[abstract:] "Taking $t$ at random, uniformly from $[0,T]$, we consider the $k$th moment, with respect to $t$, of the random variable corresponding to the $2\beta$th moment of $\zeta(1/2+ix)$ over the interval $x\in (t,t+1]$, where $\zeta(s)$ is the Riemann zeta function. We call these the `moments of moments' of the Riemann zeta function, and present a conjecture for their asymptotics, when $T\to\infty$, for integer $k$, $\beta$. This is motivated by comparisons with results for the moments of moments of the characteristic polynomials of random unitary matrices and is shown to follow from a conjecture for the shifted moments of $\zeta(s)$ due to Conrey, Farmer, Keating, Rubinstein, and Snaith. Specifically, we prove that a function which, the shifted-moment conjecture of implies, is a close approximation to the moments of moments of the zeta function does satisfy the asymptotic formula that we conjecture. We motivate as well similar conjectures for the moments of moments for other families of primitive $L$-functions."
A. Pichler, "On a rapidly converging series for the Riemann zeta function" (preprint 01/2012)
[abstract:] "To evaluate Riemann's zeta function is important for many investigations related to the area of number theory, and to have quickly converging series at hand in particular. We investigate a class of summation formulae and find, as a special case, a new proof of a rapidly converging series for the Riemann zeta function. The series converges in the entire complex plane, its rate of convergence being significantly faster than comparable representations, and so is a useful basis for evaluation algorithms. The evaluation of corresponding coefficients is not problematic, and precise convergence rates are elaborated in detail. The globally converging series obtained allow to reduce Riemann's hypothesis to similar properties on polynomials. And interestingly, Laguerre's polynomials form a kind of leitmotif through all sections."
R. Fisch, "Evidence of long range order in the Riemann zeta function" (preprint, 10/2012)
[abstract:] "We have done a statistical analysis of some properties of the contour lines $\Im(\zeta(s)) = 04 of the Riemann zeta function. We find that this function is broken up into strips whose average width on the critical line does not appear to vary with height. We also compute the position of the primary zero for the lowest 200 strips, and find that this probability distribution also appears to be scale invariant."
N. Barros e Sa and I. Bengtsson, "Families of complex Hadamard matrices" (preprint 02/2012)
[abstract:] "What is the dimension of a smooth family of complex Hadamard matrices including the Fourier matrix? We address this problem with a power series expansion. Studying all dimensions up to 100 we find that the first order result is misleading unless the dimension is 6, or a power of a prime. In general the answer depends critically on the prime number decomposition of the dimension. Our results suggest that a general theory is possible. We discuss the case of dimension 12 in detail, and argue that the solution consists of two 13-dimensional families intersecting in a previously known 9-dimensional family. A precise conjecture for all dimensions equal to a prime times another prime squared is formulated."
R.C. McPhedran, "Constructing a proof of the Riemann Hypothesis" (preprint 08/2013)
[abstract:] "This paper compares the distribution of zeros of the Riemann zeta function $\zeta(s)$ with those of a symmetric combination of zeta functions, denoted ${\cal T}_+(s)$, known to have all its zeros located on the critical line $\Re(s)=1/2$. Criteria are described for constructing a suitable quotient function of these, with properties advantageous for establishing an accessible proof that $\zeta(s)$ must also have all its zeros on the critical line: the celebrated Riemann hypothesis. While the argument put forward is not at the level of rigour required to constitute a full proof of the Riemann hypothesis, it should convince non-specialists that it must hold."
R.C. McPhedran, "The Riemann Hypothesis for Dirichlet $L$ functions" (preprint 08/2013)
[abstract:] "This paper studies the connections between the zeros and their distribution functions for two particular Dirichlet $L$ functions: the Riemann zeta function, and the Catalan beta function, also known as the Dirichlet beta function. It is shown that the Riemann hypothesis holds for the Dirichlet beta function $L_{-4}(s)$ if and only if it holds for $\zeta (s)$ - a particular case of the Generalized Riemann Hypothesis."
R.C. McPhedran and C.G. Poulton, The Riemann Hypothesis for symmetrised combinations of zeta functions" (preprint 08/2013)
[abstract:] "This paper studies combinations of the Riemann zeta function, based on one defined by P.R. Taylor, which was shown by him to have all its zeros on the critical line. With a rescaled complex argument, this is denoted here by ${\cal T}_-(s)$, and is considered together with a counterpart function ${\cal T}_+(s)$, symmetric rather than antisymmetric about the critical line. We prove that ${\cal T}_+(s)$ has all its zeros on the critical line, and that the zeros of both functions are all of first order. We establish a link between the zeros of ${\cal T}_-(s)$ and of ${\cal T}_+(s)$ with those of the zeros of the Riemann zeta function $\zeta(2 s-1)$, which enables us to prove that, if the Riemann hypothesis holds, then the distribution function of the zeros of $\zeta (2 s-1)$ agrees with those for ${\cal T}_-(s)$ and of ${\cal T}_+(s)$ in all terms which do not remain finite as $t\rightarrow \infty$."
R.C. McPhedran, L.C. Botten, D.J. Williamson, N.-A.P. Nicorovici, "The Riemann hypothesis for angular lattice sums (preprint 07/2010)
[abstract:] "We present further results on a class of sums which involve complex powers of the distance to points in a two-dimensional square lattice and trigonometric functions of their angle, supplementing those in a previous paper (McPhedran {\em et al}, 2008). We give a general expression which permits numerical evaluation of members of the class of sums to arbitrary order. We use this to illustrate numerically the properties of trajectories along which the real and imaginary parts of the sums are zero, and we show results for the first two of a particular set of angular sums (denoted ${\cal C}(1,4 m;s)$) which indicate their density of zeros on the critical line of the complex exponent is the same as that for the product (denoted ${\cal C}(0,1;s)$) of the Riemann zeta function and the Catalan beta function. We then introduce a function which is the quotient of the angular lattice sums ${\cal C}(1,4 m;s)$ with ${\cal C}(0,1;s)$, and use its properties to prove that ${\cal C}(1,4 m;s)$ obeys the Riemann hypothesis for any $m$ if and only if ${\cal C}(0,1;s)$ obeys the Riemann hypothesis. We furthermore prove that if the Riemann hypothesis holds, then ${\cal C}(1,4 m;s)$ and ${\cal C}(0,1;s)$ have the same distribution of zeros on the critical line (in a sense made precise in the proof)."
R.C. McPhedran, L.C. Botten, N.-A.P. Nicorovici, "Further results on the Riemann hypothesis for angular lattice sums" (preprint 07/2009)
[abstract:] "We present further results on a class of sums which involve complex powers of the distance to points in a two-dimensional square lattice and trigonometric functions of their angle, supplementing those in a previous paper (McPhedran et al, Proc. Roy. Soc., 2008). We give a general expression which permits numerical evaluation of members of the class of sums to arbitrary order. We use this to illustrate numerically the properties of trajectories along which the real and imaginary parts of the sums are zero, and we show results for the first two of a particular set of angular sums which indicate their density of zeros on the critical line of the complex exponent is the same as that for the product of the Riemann zeta function and the Catalan beta function."
R. C. McPhedran, "Zeros of lattice sums: 2. A geometry for the Generalised Riemann Hypothesis" (preprint 02/2016)
[abstract:] "The location of zeros of the basic double sum over the square lattice is studied. This sum can be represented in terms of the product of the Riemann zeta function and the Dirichlet beta function, so that the assertion that all its non-trivial zeros lie on the critical line is a particular case of the Generalised Riemann Hypothesis (GRH). It is shown that a new necessary and sufficient condition for this special case of the GRH to hold is that a particular set of equimodular and equiargument contours of a ratio of MacDonald function double sums intersect only on the critical line. It is further shown that these contours could only intersect off the critical line on the boundary of discrete regions of the complex plane called 'inner islands'. Numerical investigations are described related to this geometrical condition, and it is shown that for the first ten thousand zeros of both the zeta function and the beta function over 70% of zeros lie outside the inner islands, and thus would be guaranteed to lie on the critical line by the arguments presented here. A new sufficient condition for the Riemann Hypothesis to hold is also presented."
D.W. Farmer and S.M. Gonek, "Pair correlation of the zeros of the derivative of
the Riemann $\xi$-function" (preprint 03/2008)
[abstract:] "The complex zeros of the Riemann zeta-function are identical to the zeros of the Riemann xi-function, $\xi(s)$. Thus, if the
Riemann Hypothesis is true for the zeta-function, it is true for $\xi(s)$. Since $\xi(s)$ is entire, the zeros of $\xi'(s)$, its derivative, would then
also satisfy a Riemann Hypothesis. We investigate the pair correlation function of the zeros of $\xi'(s)$ under the assumption that the
Riemann Hypothesis is true. We then deduce consequences about the size of gaps between these zeros and the proportion of these zeros
that are simple."
J.B. Conrey, N.C. Snaith, "Correlations of
eigenvalues and Riemann zeros" (preprint 03/2008)
[abstract:] "We present a new approach to obtaining the lower order terms for $n$-correlation of the
zeros of the Riemann zeta function. Our approach is based on the `ratios conjecture' of Conrey, Farmer,
and Zirnbauer. Assuming the ratios conjecture we prove a formula which explicitly gives all of the lower
order terms in any order correlation. Our method works equally well for random matrix theory and gives a
new expression, which is structurally the same as that for the zeta function, for the $n$-correlation of
eigenvalues of matrices from U(N)."
E. Dueñez, D.W. Farmer, S. Froehlich, C. Hughes, F. Mezzadri and T. Phan, "Roots of the derivative of the Riemann zeta function and of characteristic polynomials" (preprint 02/2010)
[abstract:] "We investigate the horizontal distribution of zeros of the derivative of the Riemann zeta function and compare this to the radial distribution of zeros of the derivative of the characteristic polynomial of a random unitary matrix. Both cases show a surprising bimodal distribution which has yet to be explained. We show by example that the bimodality is a general phenomenon. For the unitary matrix case we prove a conjecture of Mezzadri concerning the leading order behavior, and we show that the same follows from the random matrix conjectures for the zeros of the zeta function."
S. H. Saker, "Large spaces between the zeros of the Riemann Zeta-Function" (preprint 06/09)
[abstract:] "On the hypothesis that the mixed moments of Hardy's function and its derivative are correctly predicted by random matrix theory we derive new large spaces between the zeros of the Riemann zeta-function. Our proof depends on new Wirtinger-type inequalities and numerical solutions of algebraic equations."
M. Fujimoto and K. Uehara, "A Brief note on the Riemann hypothesis" (preprint 06/09)
[abstract:] "We deal with the Euler's alternating series of the Riemann zeta function to define a regularized ratio appeared in the functional equation even in the critical strip and show some evidence to indicate the hypothesis in this note."
M. Fujimoto, K. Uehara, "A brief note on the Riemann hypothesis II" (preprint 03/2010)
[abstract:] "We deal with the Euler's alternating series of the Riemann zeta function to define a finite ratio from divergent quantities appeared in the functional equation even in the critical strip and show some evidence to indicate the hypothesis in this note."
G.K. Óttarsson, "Discrete calculus and elliptic functions" (preprint 01/2014)
G. Óttarsson,
"The Ladder Hypothesis"
[abstract:] "In a paper from 18 August 2001 available at
www.islandia.is/gko/010818.pdf,
a thermoelectric generator was constructed from a large number of series
connected parallelepipeds. The hot and/or cold reservoir was made of some
electrically conductive metal, and the fluid was to some extent conductive to the
electrical ground. This topology generated a number of small
capacitors, each formed by two parallelepiped crystal faces and the grounded
thermal reservoir. When analysing the frequency behaviour of
such a device, rational polynomials manifested themselves and proved
to be a rich source of advanced mathematical relations."
These relations involve the Riemann zeta function, Bernoulli
numbers, the Gamma function, Euler's constant and Stirling's Formula.
G. Chinta, J. Jorgenson and A. Karlsson, "Complexity and heights of tori" (preprint 10/2011)
[abstract:] "We prove detailed asymptotics for the number of spanning trees, called complexity, for a general class of discrete tori as the parameters tend to infinity. The proof uses in particular certain ideas and techniques from an earlier paper. Our asymptotic formula provides a link between the complexity of these graphs and the height of associated real tori, and allows us to deduce some corollaries on the complexity thanks to certain results from analytic number theory. In this way we obtain a conjectural relationship between complexity and regular sphere packings."
L. Vinet and H. Zhan, "Perfect state transfer on weighted graphs of the Johnson scheme" (preprint 04/2019)
[abstract:] "We characterize perfect state transfer on real-weighted graphs of the Johnson scheme $\mathcal{J}(n,k)$. Given $\mathcal{J}(n,k)=\{A_1, A_2, \cdots, A_k\}$
and $A(X) = w_0A_0 + \cdots + w_m A_m$, we show, using classical number theory
results, that $X$ has perfect state transfer at time $\tau$ if and only if
$n=2k$, $m\ge 2^{\lfloor{\log_2(k)} \rfloor}$, and there are integers $c_1,
c_2, \cdots, c_m$ such that (i) $c_j$ is odd if and only if $j$ is a power of
$2$, and (ii) for $r=1,2,\cdots,m$, \[w_r = \frac{\pi}{\tau} \sum_{j=r}^m
\frac{c_j}{\binom{2j}{j}} \binom{k-r}{j-r}.\] We then characterize perfect
state transfer on unweighted graphs of $\mathcal{J}(n,k)$. In particular, we
obtain a simple construction that generates all graphs of $\mathcal{J}(n,k)$
with perfect state transfer at time $\pi/2$."
G. Corso, "Small world effect in
natural numbers network" (preprint 09/03)
[abstract:] "We develop a network in which the natural numbers are
the vertices. We use the decomposition of natural numbers by prime
numbers to establish the connections. We perform data collapse and
show that the degree distribution of these networks scale linearly
with the number of vertices. We compare the average distance of the
network and the clustering coefficient with the distance and
clustering coefficient of the corresponding random graph. In case
we set connections among vertices each time the numbers share a
common prime number the network is not a small-world type. If the
criterium for establishing links becomes more selective, only prime
numbers greater than pl are used to establish
links, the network shows small-world effect, it means, it has high
clustering coefficient and low distance."
G. Corso, "Families
and clustering in a natural numbers network", Phys. Rev. E 69 (2004)
"We develop a network in which the natural numbers are the vertices. The decomposition
of natural numbers by prime numbers is used to establish the connections. We perform data
collapse and show that the degree distribution of these networks scales linearly with the
number of vertices. We explore the families of vertices in connection with prime numbers
decomposition. We compare the average distance of the network and the clustering
coefficient with the distance and clustering coefficient of the corresponding random
graph. In case we set connections among vertices each time the numbers share a common
prime number the network has properties similar to a random graph. If the criterion for
establishing links becomes more selective, only prime numbers greater than pl are used to
establish links, where the network has high clustering coefficient."
A.C. Kumar and S. Dasgupta, "A small world network of prime numbers",
Physica A 357 (2005) 436
[abstract:] "According to Goldbach conjecture, any even number can be broken up as the sum of two prime numbers : $n = p + q$. We
construct a network where each node is a prime number and corresponding to every even number $n$, we put a link between the component
primes $p$ and $q$. In most cases, an even number can be broken up in many ways, and then we chose {\em one} decomposition with a
probability $|p - q|^{\alpha}$. Through computation of average shortest distance and clustering coefficient, we conclude that for
$\alpha > -1.8$ the network is of small world type and for $\alpha < -1.8$ it is of regular type. We also present a theoretical
justification for such behaviour."
K.H. Knuth, "Measuring on lattices" (Presented at the 29th International Workshop on Bayesian and Maximum Entropy Methods in Science and Engineering: MaxEnt 2009)
[abstract:] "Previous derivations of the sum and product rules of probability theory relied on the algebraic properties of Boolean logic. Here they are derived within a more general framework based on lattice theory. The result is a new foundation of probability theory that encompasses and generalizes both the Cox and Kolmogorov formulations. In this picture probability is a bi-valuation defined on a lattice of statements that quantifies the degree to which one statement implies another. The sum rule is a constraint equation that ensures that valuations are assigned so as to not violate associativity of the lattice join and meet. The product rule is much more interesting in that there are actually two product rules: one is a constraint equation arises from associativity of the direct products of lattices, and the other a constraint equation derived from associativity of changes of context. The generality of this formalism enables one to derive the traditionally assumed condition of additivity in measure theory, as well introduce a general notion of product. To illustrate the generic utility of this novel lattice-theoretic foundation of measure, the sum and product rules are applied to number theory. Further application of these concepts to understand the foundation of quantum mechanics is described in a joint paper in this proceedings."
L. Alexandrov and L. Georgiev, "Prime number diffeomorphisms, Diophantine equations and the Riemann
Hypothesis" (preprint 11/04)
[abstract:] "We explicitly construct a diffeomorphic pair (p(x),p-1(x)) in terms of an
appropriate quadric spline interpolating the prime series. These continuously differentiable functions are the smooth
analogs of the prime series and the prime counting function, respectively, and contain the basic information about the
specific behavior of the primes. We employ p-1(x) to find approximate solutions of Diophantine
equations over the primes and discuss how this function could eventually be used to analyze the von Koch estimate for the
error in the prime number theorem which is known to be equivalent to the Riemann hypothesis."
J. Lagacé and L. Parovski, "A generalised Gauss circle problem and integrated density of states" (preprint 06/2015)
[abstract:] "Counting lattice points inside a ball of large radius in Euclidean space is a classical problem in analytic number theory, dating back to Gauss. We propose a variation on this problem: studying the asymptotics of the measure of an integer lattice of affine planes inside a ball. The first term is the volume of the ball; we study the size of the remainder term. While the classical problem is equivalent to counting eigenvalues of the Laplace operator on the torus, our variation corresponds to the integrated density of states of the Laplace operator on the product of a torus with Euclidean space. The asymptotics we obtain are then used to compute the density of states of the magnetic Schrödinger operator."
V. Balasubramanian, J.R. Fliss, R.G. Leigh and O. Parrikar,
Multi-boundary entanglement in Chern–Simons theory and link invariants" (preprint 11/2016)
[abstract:] "We consider Chern–Simons theory for gauge group $G$ at level $k$ on 3-manifolds $M_n$ with boundary consisting of $n$ topologically linked tori.
The Euclidean path integral on $M_n$ defines a quantum state on the boundary,
in the $n$-fold tensor product of the torus Hilbert space. We focus on the case
where $M_n$ is the link-complement of some $n$-component link inside the
three-sphere $S^3$. The entanglement entropies of the resulting states define
new, framing-independent link invariants which are sensitive to the topology of
the chosen link. For the Abelian theory at level $k$ ($G= U(1)_k$) we give a
general formula for the entanglement entropy associated to an arbitrary
$(m|n-m)$ partition of a generic $n$-component link into sub-links. The formula
involves the number of solutions to certain Diophantine equations with
coefficients related to the Gauss linking numbers (mod $k$) between the two
sublinks. This formula connects simple concepts in quantum information theory,
knot theory, and number theory, and shows that entanglement entropy between
sublinks vanishes if and only if they have zero Gauss linking (mod $k$). For $G
= SU(2)_k$, we study various two and three component links. We show that the
2-component Hopf link is maximally entangled, and hence analogous to a Bell
pair, and that the Whitehead link, which has zero Gauss linking, nevertheless
has entanglement entropy. Finally, we show that the Borromean rings have a
"W-like" entanglement structure (i.e., tracing out one torus does not lead to a
separable state), and give examples of other 3-component links which have
"GHZ-like" entanglement (i.e., tracing out one torus does lead to a separable
state)."
br>
G. Chalmers, "Data compression
with prime numbers" (preprint 11/05)
[abstract:] "A compression algorithm is presented that uses the set of prime numbers. Sequences
of numbers are correlated with the prime numbers, and labeled with the integers. The algorithm can
be iterated on data sets, generating factors of doubles on the compression."
K.K.
Nambiar, "Electrical equivalent of Riemann Hypothesis"
[abstract:] "Riemann Hypothesis is viewed as a statement about the power dissipated
in an electrical network."
P. Amore, "Convergence acceleration of series
through a variational approach" (preprint 08/04)
[abstract:] "By means of a variational approach we find new series representations both for well known mathematical
constants, such as and the Catalan constant, and for mathematical functions, such as the Riemann zeta
function. The series that we have found are all exponentially convergent and provide quite useful analytical approximations.
With limited effort our method can be applied to obtain similar exponentially convergent series for a large class of mathematical
functions."
K. Maslanka, "Hypergeometric representation of the
zeta-function of Riemann"
plus a few thoughts from a cosmologist on the nature of the zeta
function
S. Tyagi and C. Holm, "A new integral representation for the
Riemann Zeta function" (preprint 03/2007)
W. Yang, "Deligne's conjecture and mirror symmetry" (preprint 01/2020)
[abstract:] "In this paper, we will study the connections between the mirror symmetry of Calabi–Yau threefolds and Deligne's conjecture on the special values of the $L$-functions of critical motives. Using the theory of mirror symmetry, we will develop an explicit method to compute the Deligne's period for a smooth fiber in the mirror family of a one-parameter mirror pair of Calabi–Yau threefolds. We will give two classes of examples to show how this method works and express Deligne's period in terms of the classical periods of the threeform of Calabi–Yau threefolds. In the paper by Candelas, de la Ossa, Elmi and van Straten, they are able to compute the special values of the $L$-function of a Calabi–Yau threefold that is called a rank-2 attractor. In this paper, we will compute the Deligne's period of this Calabi–Yau threefold and explicitly show that it satisfies Deligne's conjecture. A second purpose of this paper is to introduce the Deligne's conjecture to the physics community, and provide further evidence that a physics theory can have applications in number theory."
C. Krattenthaler, T. Rivoal, "On
the integrality of the Taylor coefficients of mirror maps" (preprint 09/2007)
[abstract:] "We show that the Taylor coefficients of the series ${\bf q}(z)=z\exp({\bf G}(z)/{\bf F}(z))$ are
integers, where ${\bf F}(z)$ and ${\bf G}(z)+\log(z) {\bf F}(z)$ are specific solutions of certain hypergeometric
differential equations with maximal unipotent monodromy at $z=0$. We also address the question of finding the largest
integer $u$ such that the Taylor coefficients of ${\bf q}(z)^{1/u}$ are still integers. As consequences, we are able to
prove numerous integrality results for the Taylor coefficients of mirror maps of Calabi–Yau complete intersections in
weighted projective spaces, which improve and refine previous results by Lian and Yau, and by Zudilin. In particular,
we prove the general ''integrality'' conjecture of Zudilin about these mirror maps. A further outcome of the present
study is the determination of the Dwork-Kontsevich sequence $(u_N)_{N\ge1}$, where $u_N$ is the largest integer
such that $q(z)^{1/u_N}$ is a series with integer coefficients, where $q(z)=\exp(F(z)/G(z))$,
$F(z)=\sum_{m=0} ^{\infty} (Nm)! z^m/m!^N$ and $G(z)=\sum_{m=1} ^{\infty} H_{Nm}(Nm)! z^m/m!^N$,
with $H_n$ denoting the $n$-th harmonic number, conditional on the conjecture that there are no prime number $p$
and integer $N$ such that the $p$-adic valuation of $H_N$ is strictly greater than 3."
J.S. Brauchart, D.P. Hardin, E.B. Saff, "The Riesz energy
of the $N$-th roots of unity: an asymptotic expansion for large $N$" (prepring 08/2008)
[abstract:] "We derive the complete asymptotic expansion in terms of powers of $N$ for the Riesz $s$-energy of $N$
equally spaced points on the unit circle as $N\to \infty$. For $s\ge -1$, such points form optimal energy $N$-point
configurations with respect to the Riesz potential $1/r^{s}$, $s\neq0$, where $r$ is the Euclidean distance between points.
By analytic continuation we deduce the expansion for all complex values of $s$. The Riemann zeta function plays an
essential role in this asymptotic expansion."
J. Cano, M. Cheng, M. Mulligan, C. Nayak, E. Plamadeala and J. Yard, "Bulk-edge correspondence in $2+1$-dimensional Abelian topological phases" (preprint 10/2013)
[abstract:] "The same bulk two-dimensional topological phase can have multiple distinct, fully-chiral edge phases. We show that this can occur in the integer quantum Hall states at $\nu = 8$ and $12$, with experimentally-testable consequences. We show that this can occur in Abelian fractional quantum Hall states as well, with the simplest examples being at $\nu = 8/7, 12/11, 8/15, 16/5$. We give a general criterion for the existence of multiple distinct chiral edge phases for the same bulk phase and discuss experimental consequences. Edge phases correspond to lattices while bulk phases correspond to genera of lattices. Since there are typically multiple lattices in a genus, the bulk-edge correspondence is typically one-to-many; there are usually many stable fully chiral edge phases corresponding to the same bulk. We explain these correspondences using the theory of integral quadratic forms. We show that fermionic systems can have edge phases with only bosonic low-energy excitations and discuss a fermionic generalization of the relation between bulk topological spins and the central charge. The latter follows from our demonstration that every fermionic topological phase can be represented as a bosonic topological phase, together with some number of filled Landau levels. Our analysis shows that every Abelian topological phase can be decomposed into a tensor product of theories associated with prime numbers $p$ in which every quasiparticle has a topological spin that is a $p^n$-th root of unity for some $n$. It also leads to a simple demonstration that all Abelian topological phases can be represented by $U(1)^N$ Chern–Simons theory parameterized by a K-matrix."
M. Kibler, "Generalized spin bases for quantum chemistry
and quantum information" (preprint 07/2008)
[abstract:] "Symmetry adapted bases in quantum chemistry and bases adapted to quantum information share a
common characteristics: both of them are constructed from subspaces of the representation space of the group SO(3)
or its double group (i.e., spinor group) SU(2). We exploit this fact for generating spin bases of relevance for quantum
systems with cyclic symmetry and equally well for quantum information and quantum computation. Our approach is
based on the use of generalized Pauli matrices arising from a polar decomposition of SU(2). This approach leads to a
complete solution for the construction of mutually unbiased bases in the case where the dimension d of the considered
Hilbert subspace is a prime number. We also give the starting point for studying the case where d is the power of a
prime number. A connection of this work with the unitary group U(d) and the Pauli group is brielly underlined."
M.R. Kibler, "On mutually unbiased bases: Passing from $d$ to $d^2$" (preprint 10/2012)
[abstract:] "We show how to transform the problem of finding $d+1$ mutually unbiased bases in the $d$-dimensional Hilbert space into the one of finding $d(d+1)$ vectors in the $N$-dimensional Hilbert space with $N=d^2$. The transformation formulas admit a solution when $d$ is a prime number."
R. Olofsson, "Hecke eigenfunctions of quantized cat maps modulo prime powers", Annales Henri Poincaré 10 (6) (2009) 1111–1139
[abstract:] "This paper continues the work done in Olofsson [Commun. Math. Phys. 286(3):1051–1072, 2009] about the supremum norm of eigenfunctions of desymmetrized quantized cat maps. $N$ will denote the inverse of Planck's constant and we will see that the arithmetic properties of $N$ play an important role. We prove the sharp estimate $||\psi ||_{\infty} = O(N^{1/4})$ for all normalized eigenfunctions and all $N$ outside of a small exceptional set. We are also able to calculate the value of the supremum norms for most of the so called newforms. For a given $N = p^n$ , with $n>2$, the newforms can be divided in two parts (leaving out a small number of them in some cases), the first half all have supremum norm about $2/\sqrt{1\pm 1/p}$ and the supremum norm of the newforms in the second half have at most three different values, all of the order $N^1/6$. The only dependence of $A$ is that the normalization factor is different if $A$ has eigenvectors modulo $p$ or not. We also calculate the joint value distribution of the absolute value of $n$ different newforms."
J. Benatar, D. Marinucci and I. Wigman, "Planck-scale distribution of nodal length of arithmetic random waves" (preprint 11/2017)
[abstract:] "We study the nodal length of random toral Laplace eigenfunctions ("arithmetic random waves") restricted to decreasing domains ("shrinking balls"), all the way down to Planck scale. We find that, up to a natural scaling, for "generic" energies the variance of the restricted nodal length obeys the same asymptotic law as the total nodal length, and these are asymptotically fully correlated. This, among other things, allows for a statistical reconstruction of the full toral length based on partial information. One of the key novel ingredients of our work, borrowing from number theory, is the use of bounds for the so-called spectral Quasi-Correlations, i.e. unusually small sums of lattice points lying on the same circle."
W.T. Buttler, S.K. Lamoreaux and J.R. Torgerson, "A four-level single-photon quantum cryptography system based on polarization, phase and time encoding" (preprint 03/2010)
[abstract:] "We describe a quantum cryptography protocol with up to twelve four-dimensional ($d = 4$) states generated by a polarization-, phase- and time-encoding transmitter. This protocol can be experimentally realized with existing technology, drawing from time-encoded and polarization-encoded systems. The protocol is error tolerant and has a quantum bit-rate of 2 per transmission, which when combined with state detection efficiency yields a qubit efficiency of up to 1 or double that of BB84-{\it like} protocols. As a practical system, our result appears to contradict a fundamental theorem stating that there exists $d + 1$ maximally non-orthogonal bases for a $d$-dimensional space where $d$ is the power of a prime number. Evidently, this contradiction has its origin in the difference in the size of the vector spaces spanned by the basis states, semi-infinite in time and phase in our case, vs. a finite number of polarization states alone as previously considered."
S. Foldes, "The Lorentz group and its finite field
analogues: local isomorphism and approximation" (preprint 05/2008)
[abstract:] "Finite Lorentz groups acting on 4-dimensional vector spaces coordinatized by finite fields
with a prime number of elements are represented as homomorphic images of countable, rational subgroups of the
Lorentz group acting on real 4-dimensional space-time. Bounded subsets of the real Lorentz group are retractable
with arbitrary accuracy to finite subsets of such rational subgroups. These finite retracts correspond, via local
isomorphisms, to well-behaved subsets of Lorentz groups over finite fields. This establishes a relationship of
approximation between the real Lorentz group and Lorentz groups over very large finite fields."
M. Fujimoto, K. Uehara, "Regularizations of the Euler product representation for zeta
functions and the Birch-Swinnerton-Dyer conjecture" (preprint 09/2007)
[abstract:] "We consider a variant expression to regularize the Euler product representation of the zeta
functions, where we mainly apply to that of the Riemann zeta function in this paper. The regularization itself is identical
to that of the zeta function of the summation expression, but the non-use of the Möbius function enable us to confirm
a finite behavior of residual terms which means an absence of zeros except for the critical line. Same technique can be
applied to the L-function associated to the elliptic curve, and we can deal with the Taylor expansion at the pole in
critical strip which is deeply related to the Birch-Swinnerton-Dyer conjecture."
M. Fujimoto and K. Uehara, "Regularized Euler product for the zeta
function and the Birch and Swinnerton-Dyer and the Beilinson conjecture" (preprint 11/2008)
[Abstract:] "We present another expression to regularize the Euler product representation of the Riemann zeta function. In this
paper. The expression itself is essentially same as the usual Euler product that is the infinite product, but we define a new one as
the limit of the product of some terms derived from the usual Euler product. We also refer to the relation between the Bernoulli number
and $P(z)$, which is an infinite summation of a $z$ power of the inverse primes. When we apply the same technique to the $L$-function
associated to an elliptic curve, we can evaluate the power of the Taylor expansion for the function even in the critical strip, which
is deeply related to problems known as the Birch and Swinnerton-Dyer conjecture and the Beilinson conjecture."
M.W. Coffey and M.C. Lettington, "Mellin transforms with only critical zeros: Chebyshev and Gegenbauer functions" (preprint 07/2013)
[abstract:] "We consider the Mellin transforms of certain Chebyshev functions based upon the Chebyshev polynomials. We show that the transforms have polynomial factors whose zeros lie all on the critical line or on the real line. The polynomials with zeros only on the critical line are identified in terms of certain $_3F_2(1)$ hypergeometric functions. Furthermore, we extend this result to a 1-parameter family of polynomials with zeros only on the critical line. These polynomials possess the functional equation $p_n(s;\beta)=(-1)^{\lfloor n/2 \rfloor} p_n (1-s;\beta)$. We then present the generalization to the Mellin transform of certain Gegenbauer functions. The results should be of interest to special function theory, combinatorics, and analytic number theory."
M.W. Coffey and M.C. Lettington, "Mellin transforms with only critical zeros: Legendre functions" (preprint 07/2013)
[abstract:] "We consider the Mellin transforms of certain Legendre functions based upon the ordinary and associated Legendre polynomials. We show that the transforms have polynomial factors whose zeros lie all on the critical line Re $s=1/2$. The polynomials with zeros only on the critical line are identified in terms of certain $_3F_2(1)$ hypergeometric functions. These polynomials possess the functional equation $p_n(s)= (-1)^{\lfloor n/2 \rfloor} p_n(1-s)$. Other hypergeometric representations are presented, as well as certain Mellin transforms of fractional part and fractional part-integer part functions. The results should be of interest to special function theory, combinatorial geometry, and analytic number theory."
M.W. Coffey, "Series representation of the Riemann zeta function and other results: Complements to a paper of Crandall" (preprint 03/2012)
[abstract:] "We supplement a very recent paper of R. Crandall concerned with the multiprecision computation of several important special functions and numbers. We show an alternative series representation for the Riemann and Hurwitz zeta functions providing analytic continuation through out the whole complex plane. Additionally we demonstrate some series representations for the initial Stieltjes constants appearing in the Laurent expansion of the Hurwitz zeta function. A particular point of elaboration in these developments is the hypergeometric form and its equivalents for certain derivatives of the incomplete Gamma function. Finally, we evaluate certain integrals including $\int_{\tiny{Re} s=c} {{\zeta(s)} \over s} ds$ and $\int_{\tiny{Re} s=c} {{\eta(s)} \over s} ds$, with $\zeta$ the Riemann zeta function and $\eta$ its alternating form."
M.W. Coffey, "Series representations of the Riemann and Hurwitz zeta functions and series and integral representations of the first Stieltjes constant" (preprint 06/2011)
"We develop series representations for the Hurwitz and Riemann zeta functions in terms of generalized Bernoulli numbers (N\"{o}rlund polynomials), that give the analytic continuation of these functions to the entire complex plane. Special cases yield series representations of a wide variety of special functions and numbers, including log Gamma, the digamma, and polygamma functions. A further byproduct is that $\zeta(n)$ values emerge as nonlinear Euler sums in terms of generalized harmonic numbers. We additionally obtain series and integral representations of the first Stieltjes constant $\gamma_1(a)$. The presentation unifies some earlier results."
M.W. Coffey, "Evaluation of some second moment and other integrals for the Riemann, Hurwitz, and Lerch zeta functions" (preprint 02/2011)
[abstract:] "Several second moment and other integral evaluations for the Riemann zeta function $\zeta(s)$, Hurwitz zeta function $\zeta(s,a)$, and Lerch zeta function $\Phi(z,s,a)$ are presented. Additional corollaries that are obtained include previously known special cases for the Riemann zeta function $\zeta(s)=\zeta(s,1)=\Phi(1,s,1)$. An example special case is: $$\int_R {{|\zeta(1/2+it)|^2} \over {t^2+1/4}}dt=2\pi[\ln(2\pi)-\gamma],$$ with $\gamma$ the Euler constant. The asymptotic forms of certain fractional part integrals, with and without logarithmic factors in the integrand, are presented. Extensions and other approaches are mentioned."
M.W. Coffey, "Sums of alternating products of Riemann zeta values and solution of a Monthly problem" (preprint, 06/2011)
[abstract:] "We solve problem 11585 proposed by B. Burdick, AMM June-July 2011 {\bf 118} (6), p. 558 for the sum of certain products of Riemann zeta function values. We further point out an alternating sum analog, and then present and prove different alternating sum analogs. In addition, we present summation by parts and other results for the Hurwitz and Riemann zeta functions and for the digamma and trigamma functions."
M.W. Coffey, "Series of zeta values, the Stieltjes
constants, and a sum S_\gamma(n)" (preprint 06/2007)
[abstract:] "We present a variety of series representations of the Stieltjes and related constants,
the Stieltjes constants being the coefficients of the Laurent expansion of the Hurwitz zeta function
zeta(s,a) about s=1. Additionally we obtain series and integral representations of a sum S_\gamma(n)
formed as an alternating binomial series from the Stieltjes constants. The slowly varying
sum S_\gamma(n)+n is an important subsum in application of the Li criterion for the Riemann
hypothesis."
M.W. Coffey, "The Stieltjes constants, their relation
to the eta_j coefficients, and representation of the Hurwitz zeta function" (preprint 06/2007)
[abstract:] "The Stieltjes constants gamma_k(a) are the expansion coefficients in the Laurent
series for the Hurwitz zeta function about its only pole at s=1. We present the relation of
gamma_k(1) to the eta_j coefficients that appear in the Laurent expansion of the logarithmic
derivative of the Riemann zeta function about its pole at s=1. We obtain novel integral
representations of the Stieltjes constants and new decompositions such as S_2(n) = S_gamma(n) +
S_Lambda(n) for the crucial oscillatory subsum of the Li criterion for the Riemann hypothesis.
The sum S_\gamma(n) is O(n) and we present various integral representations for it. We present
novel series representations of S_2(n). We additionally present a rapidly convergent expression
for \gamma_k= \gamma_k(1) and a variety of results pertinent to a parameterized representation of
the Riemann and Hurwitz zeta functions."
Mark W. Coffey, "On representations and differences of
Stieltjes coefficients, and other relations" (preprint 09/2008)
[abstract:] "The Stieltjes coefficients $\gamma_k(a)$ arise in the expansion of the Hurwitz zeta function
$\zeta(s,a)$ about its single simple pole at $s=1$ and are of fundamental and long-standing importance in analytic
number theory and other disciplines. We present an array of exact results for the Stieltjes coefficients, including
series representations and summatory relations. Other integral representations provide the difference of Stieltjes
coefficients at rational arguments. The presentation serves to link a variety of topics in analysis and special
function and special number theory, including logarithmic series, integrals, and the derivatives of the Hurwitz zeta
and Dirichlet $L$-functions at special points. The results have a wide range of application, both theoretical and
computational."
M. Coffey, "On the coefficients of the Baez-Duarte criterion for the
Riemann hypothesis and their extensions" (preprint 08/2006)
[abstract:] "We discuss analytic properties of the constants ck appearing in the Baez-Duarte criterion for the Riemann hypothesis.
These constants are the coefficients of Pochhammer polynomials in a series representation of the reciprocal of the Riemann zeta function.
We present extensions of this representation to the Hurwitz zeta and many other special functions. We relate the corresponding
coefficients to other known constants including the Stieltjes constants and present summatory relations. In addition, we generalize the
Maslanka hypergeometric-like representation for the zeta function in several ways."
M.W. Coffey, "Special functions and the Mellin transforms of Laguerre
and Hermite functions" (preprint 12/2006)
[abstract:] "We present explicit expressions for the Mellin transforms of Laguerre and Hermite functions in terms of a variety of
special functions. We show that many of the properties of the resulting functions, including functional equations and reciprocity laws,
are direct consequences of transformation formulae of hypergeometric functions. Interest in these results is reinforced by the fact that
polynomial or other factors of the Mellin transforms have zeros only on the critical line Re s = 1/2. We additionally present a
simple-zero Proposition for the Mellin transform of the wavefunction of the D-dimensional hydrogenic atom. These results are of interest
to several areas including quantum mechanics and analytic number theory."
M.W. Coffey, "On harmonic binomial series" (preprint 12/2008)
[abstract:] "We evaluate binomial series with harmonic number coefficients, providing
recursion relations, integral representations, and several examples. The results are of
interest to analytic number theory, the analysis of algorithms, and calculations of
theoretical physics, as well as other applications."
M.W. Coffey, "Some definite logarithmic integrals from Euler sums, and other integration results" (preprint 01/2010)
[abstract:] "We present explicit expressions for multi-fold logarithmic integrals that are equivalent to sums over polygamma functions at integer argument. Such relations find application in perturbative quantum field theory, quantum chemistry, analytic number theory, and elsewhere. The analysis includes the use of properties of a variety of special functions."
S. Albino, "Analytic Continuation of
Harmonic Sums" (preprint 02/2009)
[abstract:] "We present a method for calculating any harmonic sum to arbitrary accuracy for all
complex values of the argument. The method utilizes the relation between harmonic sums and
(derivatives of) generalized Riemann zeta functions, which allows a harmonic sum to be calculated
as a Taylor series in the inverse of its argument. A program for implementing this method is also
provided."
A. Kuznetsov, "Integral representations for the
Dirichlet L-functions and their expansions in Meixner-Pollaczek polynomials and Pochhammer functions" (preprint 2006)
[abstract:] "In this article we provide integral representations for Dirichlet beta and Riemann zeta functions, which
are obtained by combining Mellin transform with fractional cosine (sine) transform. As an application of these integral
formulas we derive expansions of the above L-functions in the series of Meixner-Pollaczek polynomials and
Pochhammer functions."
M. Combescure, "The Mutually Unbiased Bases revisited" (preprint
05/2006)
[abstract:] "The study of Mutually Unbiased Bases continues to be developed vigorously, and presents several challenges in the
Quantum Information Theory. Two orthonormal bases in $\mathbb C^d, B {and} B'$ are said mutually unbiased if $\forall b\in B, b'\in B'$
the scalar product $b\cdot b'$ has modulus $d^{-1/2}$. In particular this property has been introduced in order to allow an optimization
of the measurement-driven quantum evolution process of any state $\psi \in \mathbb C^d$ when measured in the mutually unbiased bases
$B\_{j} {of} \mathbb C^d$. At present it is an open problem to find the maximal number of mutually Unbiased Bases when $d$ is not a power
of a prime number.In this article, we revisit the problem of finding Mutually Unbiased Bases (MUB's) in any dimension $d$. The
method is very elementary, using the simple unitary matrices introduced by Schwinger in 1960, together with their diagonalizations. The
Vandermonde matrix based on the $d$-th roots of unity plays a major role. This allows us to show the existence of a set of 3 MUB's in any
dimension, to give conditions for existence of more than 3 MUB's for $d$ even or odd number, and to recover the known result of existence
of $d+1$ MUB's for $d$ a prime number. Furthermore the construction of these MUB's is very explicit. As a by-product, we recover results
about Gauss Sums, known in number theory, but which have apparently not been previously derived from MUB properties."
C. Spengler and B. Kraus, "A graph state formalism for mutually unbiased bases" (preprint 09/2013)
[abstract:] "A pair of orthonormal bases is called mutually unbiased if all mutual overlaps between any element of one basis with an arbitrary element of the other basis coincide. In case the dimension, $d$, of the considered Hilbert space is a power of a prime number, complete sets of $d+1$ Mutually Unbiased Bases (MUBs) exist. Here, we present a novel method, which uses the graph state formalism to construct MUBs. We show, that for $n$ $p$-level systems, with $p$ being prime, one generalized graph state suffices to easily construct the corresponding complete set of $p^n+1$ MUBs. In fact, we show that a single $n$-dimensional vector, which is associated to the generalized graph state, can be used to generate the complete set of MUBs and demonstrate that these vectors can be easily determined."
C. Nash and D. O'Connor,
"Ray-Singer torsion, topological field theories and the Riemann zeta function
at s = 3"
C. Nash and D. O'Connor, "Determinants of Laplacians, the Ray–Singer torsion on lens spaces and the Riemann zeta function"
S. Torquato, "Basic understanding of condensed phases of matter via packing models" (preprint, 05/2018)
[abstract:] "Packing problems have been a source of fascination for millenia and their study has produced a rich literature that spans numerous disciplines. Investigations of hard-particle packing models have provided basic insights into the structure and bulk properties of condensed phases of matter, including low-temperature states (e.g., molecular and colloidal liquids, crystals and glasses), multiphase heterogeneous media, granular media, and biological systems. The densest packings are of great interest in pure mathematics, including discrete geometry and number theory. This perspective reviews pertinent theoretical and computational literature concerning the equilibrium, metastable and nonequilibrium packings of hard-particle packings in various Euclidean space dimensions. In the case of jammed packings, emphasis will be placed on the 'geometric-structure' approach, which provides a powerful and unified means to quantitatively characterize individual packings via jamming categories and 'order' maps. It incorporates extremal jammed states, including the densest packings, maximally random jammed states, and lowest-density jammed structures. Packings of identical spheres, spheres with a size distribution, and nonspherical particles are also surveyed. We close this review by identifying challenges and open questions for future research."
S. Torquato and Y. Jiao, "Dense packings of the Platonic and Archimedean solids", Nature 460 (2009) 876-879
[abstract:] "Dense packings have served as useful models of the structure of liquid, glassy and crystal states of matter, granular media, heterogeneous materials, and biological systems. Probing the symmetries and other mathematical properties of the densest packings is a problem of long-standing interest in discrete geometry and number theory. The preponderance of previous work has focused on spherical particles, and very little is known about dense polyhedral packings. We formulate the problem of generating dense packings of polyhedra within an adaptive fundamental cell subject to periodic boundary conditions as an optimization problem, which we call the Adaptive Shrinking Cell (ASC) scheme. This novel optimization problem is solved here (using a variety of multi-particle initial configurations) to find dense packings of each of the Platonic solids in three-dimensional Euclidean space. We find the densest known packings of tetrahedra, octahedra, dodecahedra and icosahedra with densities $0.782...$, $0.947...$, $0.904...$, and $0.836...$, respectively. Unlike the densest tetrahedral packing, which must be a non-Bravais lattice packing, the densest packings of the other non-tiling Platonic solids that we obtain are their previously known optimal (Bravais) lattice packings. Our simulations results, rigorous upper bounds that we derive, and theoretical arguments lead us to the strong conjecture that the densest packings of the Platonic and Archimedean solids with central symmetry are given by their corresponding densest lattice packings. This is the analog of Kepler's sphere conjecture for these solids."
S. Haran, "Index theory, potential theory, and the Riemann Hypothesis",
L-functions and Arithmetic, Durham 1990, LMS Lecture Notes
153 (1991), 257-270.
I. Fesenko, "Several nonstandard remarks"
(preprint, 2003)
[abstract:] "This text aims to present and discuss a number of situations in analysis, geometry, number
theory and mathematical physics which can profit from developing their nonstandard description or interpretation
and then using it to prove standard results and/or establish standard theories."
Section 8 concerns "Nonstandard interpretations of interactions
between noncommutative differential geometry and number theory."
[excerpt p.8-9:]
M. du Sautoy, J.
McDermott, and G. Smith, "Zeta functions of crystallographic groups and
analytic continuation", Proceedings of the London Mathematical
Society (3) 79 (1999) 511-534.
D. Bessis, J. Geronimo, and P. Moussa, "Mellin transforms associated
with Julia sets and physical applications", Journal of Statistical
Physics 34 (1984) 75-110
Xu Xinwen and Dai Xianxi, "On a specific heat-phonon spectrum
inversion problem: Exact solution, unique existence theorem and Riemann
Hypothesis", Physics Letters A 147 8,9 (1990) 445-449.
S. Garoufalidis, P. Kucharski and P. Sulkowski, "Knots, BPS states, and algebraic curves" (preprint 04/2015)
"We analyze relations between BPS degeneracies related to Labastida–Marino–Ooguri–Vafa (LMOV) invariants, and algebraic curves associated to knots. We introduce a new class of such curves that we call extremal A-polynomials, discuss their special properties, and determine exact and asymptotic formulas for the corresponding (extremal) BPS degeneracies. These formulas lead to nontrivial integrality statements in number theory, as well as to an improved integrality conjecture stronger than the known M-theory integrality predictions. Furthermore we determine the BPS degeneracies encoded in augmentation polynomials and show their consistency with known colored HOMFLY polynomials. Finally we consider refined BPS degeneracies for knots, determine them from the knowledge of super-A-polynomials, and verify their integrality. We illustrate our results with twist knots, torus knots, and various other knots with up to 10 crossings."
Y. Kano and E. Wolf, "Temporal coherence of black body radiation",
Proc. Phys. Soc. 80 (1962) 1273-1276
[abstract:] "The temporal complex coherence function of black-body radiation is calculated and is found to be expressible in terms of the
generalized Riemann $\xi$-function. Curves are given which show the variation of the absolute value and of the argument of the temporal complex
degree of coherence $\gamma(\tau)$ as functions of increasing time delay $\tau$. It is shown that the analytic continuation of $\gamma(\tau)$ has
no zeros in the lower half of the complex plane. This result supports the theory proposed in the accompanying paper by Wolf about the possibility
of determining certain energy spectra from measurements of the absolute value of the degree of coherence."
R. Marrett, "Aggregate
properties of fracture populations", Journal of Structural Geology 18 (1996) 169-178
[abstract:] "Empirical studies indicate that the individual attributes of both faults and extension fractures
follow power-law scaling. Aggregate properties of fracture populations are important in a variety of problems
and can be specified in terms of the scaling parameters of individual fracture attributes. Development of an
expression for an aggregate property requires consideration of a number of independent factors, including
the topologic dimension of the aggregate property, the topologic dimension of sampling and the possibility of
scaling changes for fractures that span some mechanically significant layer. The Riemann zeta function
provides an alternative to integration for the analytical and numerical solution of aggregate problems."
[personal communication from author, 11/07/03:] "See section on "Evaluation," pages 173-174, for the part
that might be most germane to your interests. My usage of the Riemann zeta function is shallow, really only to deal with an
infinite power series. In geology this traditionally has been done (poorly) using integration, which I show to systematically
underestimate the aggregate effects of individual features that follow a power law distibution of sizes.
I have not seen this aspect of my paper used in later publications (geologists are notoriously weak in numerical and analytical
skills). Likewise, I have not seen the Riemann zeta function used in other geological contexts; I stumbled on it in my physics
and math books while searching for a better approach to summation of infinite power series."
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 -> infinity 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."
M. Hippke, et al., "Pulsation period variations in the RRc Lyrae star KIC 5520878" (preprint 09/2014)
(Submitted on 3 Sep 2014)
"Learned et al. proposed that a sufficiently advanced extra-terrestrial civilization may tickle Cepheid and RR Lyrae variable stars with a neutrino beam at the right time, thus causing them to trigger early and jogging the otherwise very regular phase of their expansion and contraction. This would turn these stars into beacons to transmit information throughout the galaxy and beyond. The idea is to search for signs of phase modulation (in the regime of short pulse duration) and patterns, which could be indicative of intentional, omnidirectional signaling.
We have performed such a search among variable stars using photometric data from the Kepler space telescope. In the RRc Lyrae star KIC 5520878, we have found two such regimes of long and short pulse durations. The sequence of period lengths, expressed as time series data, is strongly auto correlated, with correlation coefficients of prime numbers being significantly higher (p = 99.8%). Our analysis of this candidate star shows that the prime number oddity originates from two simultaneous pulsation periods and is likely of natural origin.
Simple physical models elucidate the frequency content and asymmetries of the KIC 5520878 light curve.
Despite this SETI null result, we encourage testing other archival and future time-series photometry for signs of modulated stars. This can be done as a by-product to the standard analysis, and even partly automated."
L.B. Garrido, J. Gainza and E. Pereira, "Rheological
properties of concentrated aqueous suspensions of kaolin - application of the multiple ternary
Bingham model", Applied Clay Science 5 (1990) 217-228
[abstract:] "Several kaolin samples from Monte Salgueiro (Spain) obtained by hydrocycloning,
wet ball milling and delamination were examined. The specific lateral and basal surface areas
of the kaolinite were calculated from both the specific surface area by the BET method and the
particle size distribution curves. Flow curves of aqueous suspensions (51% w/w) of those
kaolins were obtained by a rotational viscometer of coaxial cylinders with a predetermined
history of flow. To describe the rheological behaviour the mechanical ternary Bingham model
was applied satisfactorily. Its differential equation under the movement conditions has been
integrated according to the flow history selected. The model and consequently the resulting
equation have been generalized linearly. The number of rheological parameters was reduced to
three because of the application of empirical relations among them. These parameters were
ordered according to the Riemann zeta function simplifying the calculation. Rheological
changes produced by mechanical processes can be described using the parameters of the model
which have been correlated with the geometric mean diameter, specific surface area and
cationic exchange capacity by multiple regression."
P. Exner, P. Seba, "A Markov process associated
with plot-size distribution in Czech Land Registry and its number-theoretic properties" (preprint 12/07)
[abstract:] "The size distribution of land plots is a result of land allocation processes in the past. In the absence of regulation this is a
Markov process leading an equilibrium described by a probabilistic equation used commonly in the insurance and financial mathematics. We support this claim by analyzing the distribution of two plot types, garden and build-up areas, in the Czech Land Registry pointing out the coincidence with the distribution of prime number factors described by Dickman function in the first case."
H. Frisk and S. de Gosson,
"On the motion of
zeros of zeta functions"
"The motion in the complex plane of the zeros to various zeta
functions is investigated numerically. First the Hurwitz zeta function
is considered and an accurate formula for the distribution of its
zeros is suggested. Then functions which are linear combinations of
different Hurwitz zeta functions, and have a symmetric distribution
of their zeros with respect to the critical line, are examined.
Finally the existence of the hypothetical non-trivial Riemann zeros
with Re(s) /neq 1/2 is discussed."
L. Alexandrov, "On the
nonasymptotic prime number distribution"
[abstract:] "The objective of this paper is to introduce an approach to the study of the nonasymptotic distribution
of prime numbers. The natural numbers are represented by theorem 1 in the matrix form ^2N. The first column of the
infinite matrix ^2N starts with the unit and contains all composite numbers in ascending order.The infinite rows of this
matrix except for the first elements contain prime numbers only, which are determined by an uniform recurrence law.
At least one of the elements of the twin pairs of prime numbers is an element of the second column of the matrix ^2N
(theorem 3). The basic information on the nonasymptotic prime number distribution is contained in the distribution of
the elements of the second column of the matrix ^2N."
I. Mikoss, "The prime numbers hidden
symmetric structure and its relation to the twin prime infinitude and an improved prime number theorem (preprint
2006)
[abstract:] "Due to the sieving process represented by a Secondary Sieving Map; during the generation of the
prime numbers, geometric structures with definite symmetries are formed which become evident through their
geometrical representations. The study of these structures allows the development of a constructive prime
generating formula. This defines a mean prime density yielding a second order recursive and discrete prime
producing formula and a second order differential equation whose solutions produce an improved Prime Number
Theorem. Applying these results to twin prime pairs is possible to generate a Twin Prime Number Theorem and
important conclusions about the infinitude of the twin primes."
J.S. Dowker and K. Kirsten,
"The Barnes zeta-function,
sphere determinants and Glaisher- Kinkelin-Bendersky constants"
[abstract:] "Summations and relations involving the Hurwitz and Riemann zeta-functions are
extended first to Barnes zeta-functions and then to zeta-functions of general type. The
analysis is motivated by the evaluation of determinants on spheres which are treated both by a
direct expansion method and by regularised sums. Comments on existing calculations are made. A
Kaluza-Klein technique is introduced providing a determinant interpretation of the
Glaisher-Kinkelin-Bendersky constants which are then generalised to arbitrary zeta-functions.
This technique allows an improved treatment of sphere determinants."
A. Al-Shuaibi,
"The Riemann zeta function used in the inversion of the Laplace transform",
Inverse Problems 14 (1998) 1-7.
[Abstract:] "Given the Laplace transform F(s) of a
function f(t), we develop a new algorithm to find an
approximation to f(t) by the use of the classical
Laguerre polynomials. The main contribution of our work is the
development of a new and very effective method to evaluate the
Laguerre coefficients with the use of the Riemann zeta function. Some
examples are illustrated."
D. Bryukhov, "Axially symmetric
generalization of the Cauchy-Riemann system and modified Clifford analysis" (preprint 02/03)
[abstract:] "The main aim of this paper is to describe the most adequate generalization of
the Cauchy-Riemann system fixing properties of classical functions in octonionic case. An
octonionic generalization of the Laplace transform is introduced. Octonionic generalizations
of the inversion transformation, the gamma function and the Riemann zeta-function are given."
N.V. Kuznetsov, "The
true order of the Riemann zeta-function"
"For the Riemann zeta function on the critical line the terminal
estimate has been proved, which had been conjectured by Lindelof at the
beginning of this century. The proof is based on the author's
relations which connect the bilinear forms of the eigenvalues of the Hecke
operators with sums of the Kloosterman sums."
M. Wolf, "Nearest neighbor spacing distribution of prime numbers and quantum chaos" (preprint 12/2012)
[abstract:] "We show that after appropriate rescaling the statistics of spacings between adjacent prime numbers follows the Poisson distribution. The scaling transformation removes the oscillations in the NNSD of primes. These oscillations have the very profound period of length six. We calculate the spectral rigidity $\Delta_3$ for prime numbers by two methods and we find the cross-overs in their behaviors."
G. G. Szpiro, "The
gaps between the gaps: some patterns in the prime number sequence", Physica A 341 (2004) 607-617
[abstract:] "It has long been known that the gaps between consecutive prime numbers
cluster on multiples of 6. Recently it was shown that the frequency of the gaps between
the gaps is lower for multiples of 6 than for other values (P. Kumar et. al., "Information entropy and correlation in prime numbers"). This paper
investigates "higher moments" of the prime number series and shows that they exhibit
certain peculiarities. In order to remove doubts as to whether these peculiarities are
related to the known clustering of the gaps on multiples of 6, the results are compared
to a benchmark series of "simulated gaps"."
C. Feinstein, "Complexity theory for simpletons"
(preprint, 07/2005)
[abstract:] "In this article, we shall describe some of the most interesting topics in the subject of Complexity Theory
for a general audience. Anyone with a solid foundation in high school mathematics (with some calculus) and an elementary
understanding of computer programming will be able to follow this article. First, we shall describe the P versus NP
problem and its significance. Next, we shall describe two other famous mathematics problems, the Collatz 3n+1
Conjecture and the Riemann Hypothesis, and show how the notion of "computational irreducibility" is important for
understanding why no one has, as of yet, solved these two problems."
M. Rubinstein, "Computational methods and
experiments in analytic number theory" (preprint 12/04)
[abstract:] "We cover some useful techniques in computational aspects
of analytic number theory, with specific emphasis on ideas relevant to the
evaluation of L-functions. These techniques overlap considerably with basic
methods from analytic number theory. On the elementary side, summation by
parts, Euler-Maclaurin summation, and Möbius inversion play a prominent
role. In the slightly less elementary sphere, we find tools from analysis,
such as Poisson summation, generating function methods, Cauchy's residue
theorem, asymptotic methods, and the fast Fourier transform. We then
describe conjectures and experiments that connect number theory and
random matrix theory."
P. Kurlberg and Z. Rudnick,
"The distribution of spacings between quadratic residues"
"We study the distribution of spacings between squares modulo q,
where q is square-free and highly composite, in the limit as the
number of prime factors of q goes to infinity. We show that all
correlation functions are Poissonian which among other things, implies
that the spacings between nearest neighbors, normalized to have unit mean,
have an exponential distribution."
M. Mulase, "Lectures
on the asymptotic expansion of a hermitian matrix integral"
"In these lectures three different methods of computing the asymptotic
expansion of a Hermitian matrix integral is presented... The second method
is based on the classical analysis of orthogonal polynomials. A rigorous
asymptotic method is established, and a special case of the matrix
integral is computed in terms of the Riemann zeta-function."
M.K.-H. Kiessling, "Order and chaos in some trigonometric series" (preprint 06/2012)
[abstract:] "The one-parameter family of deterministic trigonometric series $\pzcS_p: t\mapsto \sum_{n\in\Nset}\sin(n^{-{p}}t)$, $p>1$, is shown to exhibit both order and apparent chaos. It is proved that $\pzcS_p(t) = \alpha_p\rm{sign}(t)|t|^{1/{p}}+O(|t|^{1/{(p+1)}})\;\forall\;t\in\Rset$, with explicitly computed constant $\alpha_p$. A well-motivated conjecture is formulated concerning the seemingly chaotic fluctuations about this overall trend, to the effect that these fluctuations, when properly scaled, converge in distribution to a standard Gaussian when $t\to\infty$, provided that $p$ is irrational; no conjecture has been forthcoming for rational $p$. Moreover, the interesting relationship of the asymptotics of $\pzcS_p(t)$ to properties of the Riemann $\zeta$ function is worked out."
C.H.A. Cheng, R. Granero-Belinchon, S. Shkoller and J. Wilkening, "Rigorous asymptotic models of water waves" (preprint 07/2018)
"We develop a rigorous asymptotic derivation for two mathematical models of water waves that capture the full nonlinearity of the Euler equations up to quadratic and cubic interactions, respectively. Specifically, letting epsilon denote an asymptotic parameter denoting the steepness of the water wave, we use a Stokes expansion in epsilon to derive a set of linear recursion relations for the tangential component of velocity, the stream function, and the water wave parameterization. The solution of the water waves system is obtained as an infinite sum of solutions to linear problems at each $\epsilon^k$ level, and truncation of this series leads to our two asymptotic models, that we call the quadratic and cubic $h$-models. Using the growth rate of the Catalan numbers (from number theory), we prove well-posedness of the $h$-models in spaces of analytic functions, and prove error bounds for solutions of the $h$-models compared against solutions of the water waves system. We also show that the Craig–Sulem models of water waves can be obtained from our asymptotic procedure and that their WW2 model is well-posed in our functional framework. We then develop a novel numerical algorithm to solve the quadratic and cubic $h$-models as well as the full water waves system. For three very different examples, we show that the agreement between the model equations and the water waves solution is excellent, even when the wave steepness is quite large. We also present a numerical example of corner formation for water waves."
C.M. Newman and W. Wu, "Constants of de Bruijn–Newman type in analytic number theory and statistical physics" (preprint 01/2019)
[abstract:] "One formulation in 1859 of the Riemann Hypothesis (RH) was that the Fourier transform $H_f(z)$ of $f$ for $z \in \mathbb{C}$ has only real zeros when
$f(t)$ is a specific function $\Phi(t)$. Pólya's 1920s approach to RH
extended $H_f$ to $H_{f,\lambda}$, the Fourier transform of $e^{\lambda t^2}
f(t)$. We review developments of this approach to RH and related ones in
statistical physics where $f(t)$ is replaced by a measure $d \rho(t)$.
Pólya's work together with 1950 and 1976 results of de Bruijn and Newman,
respectively, imply the existence of a finite constant $\Lambda_{DN} =
\Lambda_{DN}(\Phi)$ in $(-\infty, 1/2]$ such that $H_{\Phi,\lambda}$ has only
real zeros if and only if $\lambda\geq \Lambda_{DN}$; RH is then equivalent to
$\Lambda_{DN} \leq 0$. Recent developments include the Rodgers and Tao proof of
the 1976 conjecture that $\Lamda_{DN} \geq 0$ (that RH, if true, is only
barely so) and the Polymath 15 project improving the $1/2$ upper bound to about $0.22$. We also present examples of $\rho$'s with differing $H_{\rho,\lambda}$
and $\Lambda_{DN}(\rho)$ behaviors; some of these are new and based on a
recent weak convergence theorem of the authors."
A.R. Fazely, "Prime-index parametrization for total neutrino-nucleon cross sections and $pp$ cross sections" (preprint 08/19)
"A prime number based parametrization for total neutrino-nucleon cross section
is presented. The method employs the relation between prime numbers and their
indices to reproduce neutrino cross sections for neutrino energies from the
$MeV$ to the $PeV$ regions where experimental data are available. This
prime-index relation provides estimates of the neutrino-nucleon cross sections valid across many decades of neutrino energy scales. The $PeV$ data are from the recently published astrophysical $\nu_{\mu} +\bar{\nu_{\mu}}$ rates in the IceCube detector as well as neutrino-nucleon cross section measurements. A similar method has been employed for high energy $pp$ cross sections which explains the $(\ln s)^2$ parametrization first proposed by Heisenberg."
O. Fuentealba, H.A. González, M. Pino and R. Troncoso, "The anisotropic chiral boson" (preprint 09/19)
[abstract:] "We construct the theory of a chiral boson with anisotropic scaling, characterized by a dynamical exponent $z$, whose action reduces to that of Floreanini and Jackiw in the isotropic case ($z=1$). The standard free boson with Lifshitz scaling is recovered when both chiralities are nonlocally combined. Its canonical structure and symmetries are also analyzed. As in the isotropic case, the theory is also endowed with a current algebra. Noteworthy, the standard conformal symmetry is shown to be still present, but realized in a nonlocal way. The exact form of the partition function at finite temperature is obtained from the path integral, as well as from the trace over $\hat{u}(1)$ descendants. It is essentially given by the generating function of the number of partitions of an integer into $z$-th powers, being a well-known object in number theory. Thus, the asymptotic growth of the number of states at fixed energy, including subleading corrections, can be obtained from the appropriate extension of the renowned result of Hardy and Ramanujan."
L.A. Takhtajan, "Etudes of the resolvent" (preprint 04/2020)
[abstract:] "Based on the notion of the resolvent and on the Hilbert identities, this paper presents a number of classical results in the theory of differential operators and some of their applications to the theory of automorphic functions and number theory from a unified point of view. For instance, for the Sturm–Liouville operator there is a derivation of the Gelfand–Levitan trace formula, and for the one-dimensional Schroedinger operator a derivation of Faddeev's formula for the characteristic determinant and the Zakharov–Faddeev trace identities. Recent results on the spectral theory of a certain functional-difference operator arising in conformal field theory are then presented. The last section of the survey is devoted to the Laplace operator on a fundamental domain of a Fuchsian group of the first kind on the Lobachevsky plane. An algebraic scheme is given for proving analytic continuation of the integral kernel of the resolvent of the Laplace operator and the Eisenstein–Maass series. In conclusion, there is a discussion of the relation between the values of the Eisenstein–Maass series at Heegner points and Dedekind zeta-functions of imaginary quadratic fields, and it is explained why pseudo-cuspforms for the case of the modular group do not provide any information about the zeros of the Riemann zeta-function."
B. Mukhametzhanov and S. Pal, "Beurling–Selberg extremization and modular bootstrap at high energies" (preprint 03/2020)
[abstract:] "We consider previously derived upper and lower bounds on the number of operators in a window of scaling dimensions $[\Delta-\delta,\Delta+\delta]$ at asymptotically large $\Delta$ in $2d$ unitary modular invariant CFTs. These bounds depend on a choice of functions that majorize and minorize the characteristic function of the interval $[\Delta-\delta,\Delta+\delta]$ and have Fourier transforms of finite support. The optimization of the bounds over this choice turns out to be exactly the Beurling–Selberg extremization problem, widely known in analytic number theory. We review solutions of this problem and present the corresponding bounds on the number of operators for any $\delta\geq 0$. When $2\delta \in \mathbb{Z}_{\geq 0}$ the bounds are saturated by known partition functions with integer-spaced spectra. Similar results apply to operators of fixed spin and Virasoro primaries in $c > 1$ theories."
P. Dutta and D. Ghoshal, "Pseudodifferential operators on $\mathbf{Q}_p$ and $L$-series" (preprint 02/2020)
[abstract:] "We define a family of pseudodifferential operators on the Hilbert space
$L^2(\mathbf{Q}_p)$ of complex valued square-integrable functions on the $p$-adic number field $\mathbf{Q}_p$. The Riemann zeta-function and the related Dirichlet $L$-functions can be expressed as a trace of these operators on a subspace of $L^2(\mathbf{Q}_p)$. We also extend this to the $L$-functions associated with modular (cusp) forms. Wavelets on $L^2(\mathbf{Q}_p)$ are common sets of eigenfunctions of these operators."
M. Kovačević, "Signaling to relativistic observers: An Einstein–Shannon–Riemann encounter" (preprint 05/2020)
[abstract:] "A communication scenario is described involving a series of events triggered by a transmitter and observed by a receiver experiencing relativistic time dilation. The message selected by the transmitter is assumed to be encoded in the events' timings and is required to be perfectly recovered by the receiver, regardless of the difference in clock rates in the two frames of reference. It is shown that the largest proportion of the space of all $k$-event signals that can be selected as a code ensuring error-free information transfer in this setting equals $\zeta(k)^{−1}$, where $\zeta$ is the Riemann zeta function."
A. T. DeCelles, "Global automorphic Sobolev theory and the automorphic heat kernel" (preprint 02/2021)
[abstract:] "Heat kernels arise in a variety of contexts including probability, geometry, and functional analysis; the automorphic heat kernel is particularly important in number theory and string theory. The typical construction of an automorphic heat kernel as a Poincaré series presents analytic difficulties, which can be dealt with in special cases (e.g. hyperbolic spaces) but are often sidestepped in higher rank by restricting to the compact quotient case. In this paper, we present a new approach, using global automorphic Sobolev theory, a robust framework for solving automorphic PDEs that does not require any simplifying assumptions about the rank of the symmetric space or the compactness of the arithmetic quotient. We construct an automorphic heat kernel via its automorphic spectral expansion in terms of cusp forms, Eisenstein series, and residues of Eisenstein series. We then prove uniqueness of the automorphic heat kernel as an application of operator semigroup theory. Finally, we prove the smoothness of the automorphic heat kernel by proving that its automorphic spectral expansion converges in the $C^{\infty}$-topology."
P. Flajolet and I. Vardi, "Zeta function expansions
of classical constants" (preprint, 1996)
Frantisek Slanina's page on "Number theory & informatics & codes"
"Langlands on Langlands" and
"More On Geometric Langlands (a Grand Unified Theory of Math?)"
(articles from Peter Woit's blog Not Even Wrong)
Wikipedia article on Langlands Program
Wikipedia article on Robert Langlands
Workgroup: Zeta
functions and locally symmetric spaces
(Technical University, Clausthal, Germany)
Harmonic analysis and zeta functions
Thermodynamic formalisms and dynamic zeta functions
Arithmetic theory of zeta functions
A.M. Selvam,
"Quantum-like chaos in prime number distribution and in turbulent fluid flows"
"The model concepts enable to show that the continuum real number field
contains unique structures, namely prime numbers which are analogous
to the dominant eddies in the eddy continuum in turbulent fluid flows. In
this paper it is shown that the prime number frequency spectrum follows
quantumlike mechanical laws."
This is an elaboration of the following, earlier version:
A.M. Selvam,
"Cantorian fractal patterns, quantum-like chaos and prime numbers in
atmospheric flows"
"The quantum-like chaos in atmospheric flows can be quantified in terms
of the following mathematical functions/concepts: (1) The fractal structure
of the flow pattern is resolved into an overall logarithmic spiral trajectory with the quasiperiodic Penrose tiling pattern for the
internal structure and is equivalent to a hierarchy of vortices. The
incorporation of Fibonacci mathematical series, representative of ramified
bifurcations, indicates ordered growth of fractal patterns. (2) The steady
state emergence of progressively larger fractal structures incorporates
unique primary perturbation domains of progressively increasing number
equal to z/ln z where z, the length step growth stage
is equal to the length scale ratio of large eddy to turbulent eddy. In
number theory, z/ln z gives the number of primes less than
z. The model also predicts that z/ln z represents the
normalised cumulative variance spectrum of the eddies and which follows
statistical normal distribution. The important result of the study is that
the prime number spectrum is the same as the eddy energy spectrum for
quantum-like chaos in atmospheric flows."
A.M. Selvam,
"Signatures of quantum-like chaos in spacing intervals of non-trivial
Riemann zeta zeros and in turbulent fluid flows"
The following article is unrelated to number theory, but provided a
conceptual framework around which the above articles were constructed.
A.M. Selvam,
"Universal quantification for deterministic chaos in dynamical
systems"
Gary W. Adamson has supplied four new entries to
Sloane's Encyclopedia of Integer
Sequences, relating to the "heights" of the
Riemann zeros. Access the search field at the website using "sequence number", and the following: A100060, A102522,
A102523, A102524.
the Ulam spiral phenomenon
books
L. Nyssen (ed.), Physics and
Number Theory, IRMA Lectures in Mathematics and Theoretical Physics, vol. 10 (European Mathematics
Society, 2006)
[publisher's description:] "There is a rich and historical relationship between theoretical physics and number
theory. This volume presents a selection of problems which are currently in full development and inspire a lot of
research going on. Each of the seven contributions starts with an introductory survey which makes it possible even
for non-specialists to understand the results and to gain an idea of the great variety of subjects and techniques
used.
Topics covered are: phase locking in oscillating systems, crystallography, Hopf algebras and renormalisation
theory, Zeta-function and random matrices, Kloosterman sums and the local Langlands correspondence.
Intended for research mathematicians and theoretical physicists as well as graduate students, this volume gives an
overview of recent developments in an exciting subject crossing several disciplines."
P.E. Cartier, B. Julia, P. Moussa and P. Vanhove (eds.), Frontiers in Number Theory,
Physics, and Geometry: On Random Matrices, Zeta Functions, and Dynamical Systems (Springer, due March 2006)
[publisher's description:] "This book presents pedagogical contributions on selected topics relating Number Theory, Theoretical
Physics and Geometry. The parts are composed of long self-contained pedagogical lectures followed by shorter contributions
on specific subjects organized by theme. Most courses and short contributions go up to the recent developments in the fields;
some of them follow their author's original viewpoints. There are contributions on Random Matrix Theory, Quantum Chaos,
Non-commutative Geometry, Zeta functions, and Dynamical Systems. The chapters of this book are extended versions of
lectures given at a meeting entitled Number Theory, Physics and Geometry,
held at Les Houches in March 2003, which gathered mathematicians and physicists."
Number Theory and Physics, M.
Waldschmidt, et al., eds. Proceedings of the Winter School, Les
Houches, France, March 7–16, 1989, Springer Proceedings in Physics 47
(Springer, 1989)
From Number Theory to Physics, M.
Waldschmidt, et al., eds. Further proceedings of the winter school,
Les Houches, France, March 7–16, 1989 (Springer, 1992)
S.A. Burr, G.E. Andrews (editors),
The Unreasonable Effectiveness of Number Theory (Proceedings
of Symposia in Applied Mathematics, Vol. 46) (AMS, 1992).
D. Hejhal, et. al., eds., Emerging Applications of
Number Theory
, (Springer, 1999)