## recently archived material[Items are added to the top of this list as they are archived elsewhere.]

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

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

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

F. Murtagh, Hierarchical matching and regression with application to photometric redshift estimation" (preprint, 12/2016)

[abstract:] "This work emphasizes that heterogeneity, diversity, discontinuity, and discreteness in data is to be exploited in classification and regression problems. A global a priori model may not be desirable. For data analytics in cosmology, this is motivated by the variety of cosmological objects such as elliptical, spiral, active, and merging galaxies at a wide range of redshifts. Our aim is matching and similarity-based analytics that takes account of discrete relationships in the data. The information structure of the data is represented by a hierarchy or tree where the branch structure, rather than just the proximity, is important. The representation is related to p-adic number theory. The clustering or binning of the data values, related to the precision of the measurements, has a central role in this methodology. If used for regression, our approach is a method of cluster-wise regression, generalizing nearest neighbour regression. Both to exemplify this analytics approach, and to demonstrate computational benefits, we address the well-known photometric redshift or 'photo-z' problem, seeking to match Sloan Digital Sky Survey (SDSS) spectroscopic and photometric redshifts."

R. Sasaki, Symmetric Morse potential is exactly solvable" (preprint, 11/2016)

[abstract:] "Morse potential $V_M(x)= g^2\exp (2x)-g(2h+1)\exp(x)$ is defined on the full line, $-\infty<x<\infty$ and it defines an exactly solvable 1-d quantum mechanical system with finitely many discrete eigenstates. By taking its right half $0\le x<\infty$ and glueing it with the left half of its mirror image $V_M(-x)$, $-\infty<x\le0$, the symmetric Morse potential $V(x)= g^2\exp (2|x|)-g(2h+1)\exp(|x|)$ is obtained. The quantum mechanical system of this piecewise analytic potential has infinitely many discrete eigenstates with the corresponding eigenfunctions given by the Whittaker W function. The eigenvalues are the square of the zeros of the Whittaker function $W_{k,\nu}(x)$ and its linear combination with $W'_{k,\nu}(x)$ as a function of $\nu$ with fixed $k$ and $x$. This quantum mechanical system seems to offer an interesting example for discussing the Hilbert--P\'olya conjecture on the pure imaginary zeros of Riemann zeta function on Re$(s)=\tfrac12$."

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)."

J. Berra-Montiel and A. Molgado, "Polymeric quantum mechanics and the zeros of the Riemann zeta function" (preprint, 10/2016)

[abstract:] "We analyze the Berry–Keating model and the Sierra and Rodríguez–Laguna Hamiltonian within the polymeric quantization formalism. By using the polymer representation, we obtain for both models, the associated polymeric quantum Hamiltonians and the corresponding stationary wave functions. The self-adjointness condition provide a proper domain for the Hamiltonian operator and the energy spectrum, which turned out to be dependent on an introduced scale parameter. By performing a counting of semiclassical states, we prove that the polymer representation reproduces the smooth part of the Riemann–von Mangoldt formula, and introduces a correction depending on the energy and the scale parameter, which resembles the fluctuation behavior of the Riemann zeros."

L. Albert and M. K.-H. Kiessling, "Order and Chaos in some deterministic infinite trigonometric products" (preprint, 09/2016)

[abstract:] "In this paper it is proved that $\prod_{n=1}^\infty \left[\frac23+\frac13\cos\left(\frac{x}{n^{2}}\right)\right] = e^{- C \,\sqrt{|x|} +\varepsilon(|x|)},$ with $|\varepsilon(|x|)| \leq K |x|^{1/3}$ for some $K>0$, and with $C= \int\frac{\sin\xi^2}{2+\cos\xi^2}{\rm{d}}\xi;$ numerically, $C = 0.319905585... \sqrt{\pi}$. As a corollary this confirms a surmise of Benoit Cloitre. The $O\big(|x|^{1/3}\big)$ error bound is empirically found to be accurate for moderately sized $|x|$ but not for larger $|x|$. This difference $\varepsilon(|x|)$ between Cloitre's $\log \prod_{n\geq1}\left[\frac23 +\frac13\cos\left(\frac{x}{n^{2}}\right)\right]$ and its regular trend $-C\sqrt{|x|}$, although deterministic, appears to be an "empirically unpredictable" function. A probabilistic investigation of this phenomenon is carried out, proving that Cloitre's trigonometric product is the characteristic function of a simple random walk on the real interval $(-\zeta(2),\zeta(2))$, where $\zeta$ is Riemann's zeta function, in fact, this random walk is a 'random Riemann-$\zeta$ function with argument 2.' A few related random walks are studied empirically and compared with their theoretical distributions and trend distributions. The paper closes with a paradoxical random-walk scenario and a remark on the Riemann hypothesis."

[abstract:] "Although the notion of superdeterminism can, in principle, account for the violation of the Bell inequalities, this potential explanation has been roundly rejected by the quantum foundations community. The arguments for rejection, one of the most substantive coming from Bell himself, are critically reviewed. In particular, analysis of Bell's argument reveals an implicit unwarranted assumption: that the Euclidean metric is the appropriate yardstick for measuring distances in state space. Bell's argument is largely negated if this yardstick is instead based on the alternative p-adic metric. Such a metric, common in number theory, arises naturally when describing chaotic systems which evolve precisely on self-similar invariant sets in their state space. A locally-causal realistic model of quantum entanglement is developed, based on the premise that the laws of physics ultimately derive from an invariant-set geometry in the state space of a deterministic quasi-cyclic mono-universe. Based on this, the notion of a complex Hilbert vector is reinterpreted in terms of an uncertain selection from a finite sample space of states, leading to a novel form of 'consistent histories' based on number-theoretic properties of the transcendental cosine function. This leads to novel realistic interpretations of position/momentum non-commutativity, EPR, the Bell Theorem and the Tsirelson bound. In this inherently holistic theory – neither conspiratorial, retrocausal, fine tuned nor nonlocal – superdeterminism is not invoked by fiat but is emergent from these 'consistent histories' number-theoretic constraints. Invariant set theory provides new perspectives on many of the contemporary problems at the interface of quantum and gravitational physics, and, if correct, may signal the end of particle physics beyond the Standard Model."

A. Sowa, "Encoding spatial data into quantum observables" (preprint 09/2016)

[abstract:] "The focus of this work is a correspondence between the Hilbert space operators on one hand, and doubly periodic generalized functions on the other. The linear map that implements it, referred to as the Q-transform, enables a direct application of the classical Harmonic Analysis in a study of quantum systems. In particular, the Q-transform makes it possible to reinterpret the dynamic of a quantum observable as a (typically nonlocal) dynamic of a classical observable. From this point of view we carry out an analysis of an open quantum system whose dynamics are governed by an asymptotically harmonic Hamiltonian and compact type Lindblad operators. It is established that the initial value problem of the equivalent nonlocal but classical evolution is well posed in the appropriately chosen Sobolev spaces. The second set of results pertains to a generalization of the basic Q-transform and highlights a certain type of asymptotic redundancy. This phenomenon, referred to as the broadband redundancy, is a consequence of a well-known property of the zeros of the Riemann zeta function, namely, the uniform distribution modulo one of their ordinates. Its relevance to the analysis of quantum dynamics is only a special instance of its utility in harmonic analysis in general. It remains to be seen if the phenomenon is significant also in the physical sense, but it appears well-justified—in particular, by the results presented here—to pose such a question."

G. Cotti, "Coalescence phenomenon of quantum cohomology of Grassmannians and the distribution of prime numbers" (preprint 08/2016)

[abstract:] "The occurrence and frequency of a phenomenon of resonance (namely the coalescence of some Dubrovin canonical coordinates) in the locus of Small Quantum Cohomology of complex Grassmannians is studied. It is shown that surprisingly this frequency is strictly subordinate and highly influenced by the distribution of prime numbers. Two equivalent formulations of the Riemann Hypothesis are given in terms of numbers of complex Grassmannians without coalescence: the former as a constraint on the disposition of singularities of the analytic continuation of the Dirichlet series associated to the sequence counting non-coalescing Grassmannians, the latter as asymptotic estimate (whose error term cannot be improved) for their distribution function."

V. K. Varma, S. Pilati and V. E. Kravtsov, "Conduction in quasi-periodic and quasi-random lattices: Fibonacci, Riemann, and Anderson models" (preprint 07/2016)

[abstract:] "We study the ground state conduction properties of noninteracting electrons in aperiodic but non-random one-dimensional models with chiral symmetry, and make comparisons against Anderson models with non-deterministic disorder. The first model we consider is the Fibonacci lattice, which is a paradigmatic model of quasicrystals; the second is the Riemann lattice, which we define inspired by Dyson's proposal on the possible connection between the Riemann hypothesis and a suitably defined quasicrystal. Our analysis is based on Kohn's many-particle localization tensor defined within the modern theory of the insulating state. In the Fibonacci quasicrystal, where all single-particle eigenstates are critical (i.e., intermediate between ergodic and localized), the noninteracting electron gas is found to be a conductor at most electron densities, including the half-filled case; however, at various specific fillings $\rho$, including the values $\rho = 1/\rho^n$, where $g$ is the golden ratio and $n$ is any integer, the gas turns into an insulator due to spectral gaps. Metallic behaviour is found at half-filling in the Riemann lattice as well; however, in contrast to the Fibonacci quasicrystal, the Riemann lattice is generically an insulator due to single-particle eigenstate localization, likely at all other fillings. Its behaviour turns out to be alike that of the off-diagonal Anderson model, albeit with different system-size scaling of the band-centre anomalies. The advantages of analysing the Kohn's localization tensor instead of other measures of localization familiar from the theory of Anderson insulators (such as the participation ratio or the Lyapunov exponent) are highlighted."

R. Ramanathan, M. Túlio Quintino, A. Belén Sainz, G. Murta, R. Augusiak, "On the tightness of correlation inequalities with no quantum violation" (preprint 07/2016)

[abstract:] "We study the faces of the set of quantum correlations, i.e., the Bell and noncontextuality inequalities without any quantum violation. First, we investigate the question whether every proper (tight) Bell inequality for two parties, other than the trivial ones from positivity, normalization and no-signaling can be violated by quantum correlations, i.e., whether the classical Bell polytope or the smaller correlation polytope share any facets with their respective quantum sets. To do this, we develop a recently derived bound on the quantum value of linear games based on the norms of game matrices to give a simple sufficient condition to identify linear games with no quantum advantage. Additionally we show how this bound can be extended to the general class of unique games, illustrating it for the case of three outcomes. We then show as a main result that the paradigmatic examples of correlation Bell inequalities with no quantum violation, namely the non-local computation games do not constitute tight Bell inequalities, not even for the correlation polytope. We also extend this to an arbitrary prime number of outcomes for a specific class of these games. We then study the faces in the simplest CHSH Bell scenario of binary dichotomic measurements, and identify edges in the set of quantum correlations in this scenario.

Finally, we relate the non-contextual polytope of single-party correlation inequalities with the cut polytope $CUT(\nabla G)$, where $G$ denotes the compatibility graph of observables in the contextuality scenario and $\nabla G$ denotes the suspension graph of $G$. We observe that there exist tight non-contextuality inequalities with no quantum violation, and furthermore that this set of inequalities is beyond those implied by the Consistent Exclusivity principle."

C.M. Bender, D.C. Brody and M.P. Müller, "Hamiltonian for the zeros of the Riemann zeta function" (preprint 08/2016)

[abstract:] "A Hamiltonian operator $\^H$ is constructed with the property that if the eigenfunctions are formally required to obey a suitable boundary condition, then the associated eigenvalues correspond to the nontrivial zeros of the Riemann zeta function. The classical limit of $\^H$ is $2xp$, which is consistent with the Berry–Keating conjecture. While $\^H$ is not Hermitian, $i\^H$ is symmetric under $\mathcal{PT}$ reflection. The $\mathcal{PT}$ symmetry appears to be maximally broken, which suggests that the eigenvalues of $\^H$ are real and that the Riemann hypothesis is valid. Further heuristic analysis of $\^H$ supports the conclusion that its eigenvalues are real."

F. Bornemann, P.J. Forrester and A. Mays, "Finite size effects for spacing distributions in random matrix theory: Circular ensembles and Riemann zeros" (preprint 08/2016)

[abstract:] "According to Dyson's three fold way, from the viewpoint of global time reversal symmetry there are three circular ensembles of unitary random matrices relevant to the study of chaotic spectra in quantum mechanics. These are the circular orthogonal, unitary and symplectic ensembles, denoted COE, CUE and CSE respectively. For each of these three ensembles and their thinned versions, whereby each eigenvalue is deleted independently with probability $1 - \xi$, we take up the problem of calculating the first two terms in the scaled large $N$ expansion of the spacing distributions. It is well known that the leading term admits a characterisation in terms of both Fredholm determinants and Painlevé transcendents. We show that modifications of these characterisations also remain valid for the next to leading term, and that they provide schemes for high precision numerical computations. In the case of the CUE there is an application to the analysis of Odlyzko's data set for the Riemann zeros, and in that case some further statistics are similarly analysed."

K.V. Shajesh, I. Brevik, I. Cavero-Peláez and P. Parashar, "Self-similar plates: Casimir energies" (preprint 07/2016)

[abstract:] "We construct various self-similar configurations using parallel $\delta$-function plates and show that it is possible to evaluate the Casimir interaction energy of these configurations using the idea of self-similarity alone. We restrict our analysis to interactions mediated by a scalar field, but the extension to electromagnetic field is immediate. Our work unveils an easy and powerful method that can be easily employed to calculate the Casimir energies of a class of self-similar configurations. As a highlight, in an example, we determine the Casimir interaction energy of a stack of parallel plates constructed by positioning $\delta$-function plates at the points constituting the Cantor set, a prototype of a fractal. This, to our knowledge, is the first time that the Casimir energy of a fractal configuration has been reported. Remarkably, the Casimir energy of some of the configurations we consider turn out to be positive, and a few even have zero Casimir energy. For the case of positive Casimir energy that is monotonically decreasing as the stacking parameter increases the interpretation is that the pressure of vacuum tends to inflate the infinite stack of plates. We further support our results, derived using the idea of self-similarity alone, by rederiving them using the Green's function formalism. These expositions gives us insight into the connections between the regularization methods used in quantum field theories and regularized sums of divergent series in number theory."

I. Gálvez-Carrillo, R.M. Kaufmann and A. Tonks, "Three Hopf algebras and their common simplicial and categorical background" (preprint 07/2016)

[abstract:] "We consider three a priori totally different setups for Hopf algebras from number theory, mathematical physics and algebraic topology. These are the Hopf algebras of Goncharov for multiple zeta values, that of Connes–Kreimer for renormalization, and a Hopf algebra constructed by Baues to study double loop spaces. We show that these examples can be successively unified by considering simplicial objects, cooperads with multiplication and Feynman categories at the ultimate level. These considerations open the door to new constructions and reinterpretation of known constructions in a large common framework."

T. Olupitan, C. Lei and A. Vourdas, "An analytic function approach to weak mutually unbiased bases" (preprint 07/2016) "Quantum systems with variables in $\mathbb{Z}(d)$ are considered, and three different structures are studied. The first is weak mutually unbiased bases, for which the absolute value of the overlap of any two vectors in two different bases is $1/\sqrt{k}$ (where $k\vert d$) or $0$. The second is maximal lines through the origin in the $\mathbb{Z}(d)\times \mathbb{Z}(d)$ phase space. The third is an analytic representation in the complex plane based on Theta functions, and their zeros. It is shown that there is a correspondence (triality) that links strongly these three apparently different structures. For simplicity, the case where $d = p_1\times p_2$, where $p_1$, $p_2$ are odd prime numbers different from each other, is considered."

D. Broadhurst, "Feynman integrals, $L$-series and Kloosterman moments" (preprint 02/2016)

[abstract:] "This work lies at an intersection of three subjects: quantum field theory, algebraic geometry and number theory, in a situation where dialogue between practitioners has revealed rich structure. It contains a theorem and 7 conjectures, tested deeply by 3 optimized algorithms, on relations between Feynman integrals and $L$-series defined by products, over the primes, of data determined by moments of Kloosterman sums in finite fields. There is an extended introduction, for readers who may not be familiar with all three of these subjects. Notable new results include conjectural evaluations of non-critical $L$-series of modular forms of weights 3, 4 and 6, by determinants of Feynman integrals, an evaluation for the weight 5 problem, at a critical integer, and formulas for determinants of arbitrary size, tested up to 30 loops. It is shown that the functional equation for Kloosterman moments determines much but not all of the structure of the $L$-series. In particular, for problems with odd numbers of Bessel functions, it misses a crucial feature captured in this work by novel and intensively tested conjectures. For the 9-Bessel problem, these lead to an astounding compression of data at the primes."

O. Barrett, P. Burkhardt, J. DeWitt, R. Dorward and S.J. Miller, "One-level density for holomorphic cusp forms of arbitrary level" (preprint 04/2016)

"In 2000 Iwaniec, Luo, and Sarnak proved for certain families of $L$-functions associated to holomorphic newforms of square-free level that, under the Generalized Riemann Hypothesis, as the conductors tend to infinity the one-level density of their zeros matches the one-level density of eigenvalues of large random matrices from certain classical compact groups in the appropriate scaling limit. We remove the square-free restriction by obtaining a trace formula for arbitrary level by using a basis developed by Blomer and Milicevic, which is of use for other problems as well."

V. Blomer, J. Bourgain and Z. Rudnick, "Small gaps in the spectrum of the rectangular billiard" (preprint 04/2016)

"We study the size of the minimal gap between the first $N$ eigenvalues of the Laplacian on a rectangular billiard having irrational squared aspect ratio $\alpha$, in comparison to the corresponding quantity for a Poissonian sequence. If $\alpha$ is a quadratic irrationality of certain type, such as the square root of a rational number, we show that the minimal gap is roughly of size $1/N$, which is essentially consistent with Poisson statistics. We also give related results for a set of $\alpha$'s of full measure. However, on a fine scale we show that Poisson statistics is violated for all $\alpha$. The proofs use a variety of ideas of an arithmetical nature, involving Diophantine approximation, the theory of continued fractions, and results in analytic number theory. One of our results is conditional on the Riemann Hypothesis."

F. Qiu, "A necessary condition for the existence of the nontrivial zeros of the Riemann zeta function" (preprint 01/2008)

[abstract:] "Starting from the symmetrical reflection functional equation of the zeta function, we have found that the $\sigma$ values satisfying $\zeta(s) = 0$ must also satisfy $|\zeta(s)| = |\zeta(1 - s)|$. The $\sigma$ values satisfying this requirement are a necessary condition for the existence of the nontrivial zeros. We have shown that $\sigma = 1/2$ is the only numeric solution that satisfies the requirement."

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."

G. A. P. Ribeiro and A. Klümper, "Correlation functions of the integrable spin-$s$ chain" (preprint 02/2016)

[abstract:] "We study the correlation functions of $\mathrm{SU}(2)$ invariant spin-$s$ chains in the thermodynamic limit. We derive non-linear integral equations for an auxiliary correlation function $\omega$ for any spin $s$ and finite temperature $T$. For the spin-$3/2$ chain for arbitrary temperature and zero magnetic field we obtain algebraic expressions for the reduced density matrix of two-sites. In the zero temperature limit, the density matrix elements are evaluated analytically and appear to be given in terms of Riemann's zeta function values of even and odd arguments."

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

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

J.L. Rosales and V. Martin, "On the quantum simulation of the factorization problem" (preprint 01/2016)

[abstract:] "Feynman's prescription for a quantum computer was to find a Hamitonian for a system that could serve as a computer. Here we concentrate in a system to solve the problem of decomposing a large number $N$ into its prime factors. The spectrum of this computer is exactly calculated obtaining the factors of $N$ from the arithmetic function that represents the energy of the computer.

As a corollary, in the semi-classical large $N$ limit, we compute a new prime counting asymptote $\pi(x|N)$, where $x$ is a candidate to factorize $N$, that has no counterpart in analytic number theory. This rises the conjecture that the quantum solution of factoring obtains prime numbers, thus reaching consistency with Euclid's unique factorization theorem: primes should be quantum numbers of a Feynman's factoring simulator."

J. Ryu, M. Marciniak, M. Wiesniak and M. Zukowski, "Entanglement conditions for integrated-optics multi-port quantum interferometry experiments" (preprint 01/2016)

[abstract:] "Integrated optics allows one to perform interferometric experiments based upon multi-port beam-splitter. To observe entanglement effects one can use multi-mode parametric down-conversion emissions. When the structure of the Hamiltonian governing the emissions has (infinitely) many equivalent Schmidt decompositions into modes (beams), one can have perfect EPR-like correlations of numbers of photons emitted into "conjugate modes" which can be monitored at spatially separated detection stations. We provide series of entanglement conditions for all prime numbers of modes, and show their violations by bright multi-mode squeezed vacuum states. One family of such conditions is given in terms of the usual intensity-related variables. Moreover, we show that an alternative series of conditions expressed in terms averages of observed rates, which is a generalization of the ones given in arXiv:1508.02368, is a much better entanglement indicator. Thus the rates seem to emerge as a powerful concept in quantum optics. Generalizations of the approach are expected."

G. Sierra, "The Riemann zeros as spectrum and the Riemann hypothesis" (preprint 01/2016)

"We review a series of works whose aim is to provide a spectral realization of the Riemann zeros and that culminate in a physicist's proof of the Riemann hypothesis. These results are obtained analyzing the spectrum of the Hamiltonian of a massless Dirac fermion in a region of Rindler spacetime that contains moving mirrors whose accelerations are related to the prime numbers. We show that a zero on the critical line becomes an eigenvalue of the Hamiltonian in the limit where the mirrors become transparent, and the self-adjoint extension of the Hamiltonian is adjusted accordingly with the phase of the zeta function. We have also considered the spectral realization of zeros off the critical line using a non self-adjoint operator, but its properties imply that those zeros do not exist. In the derivation of these results we made several assumptions that need to be established more rigorously."

P.A.M. Dirac, "The relation between mathematics and physics" (lecture delivered on presentation of the James Scott prize, 6 February 1939)

[excerpt:] "There is thus a possibility that the ancient dream of philosophers to connect all Nature with the properties of whole numbers will some day be realized. To do so physics will have to develop a long way to establish the details of how the correspondence is to be made. One hint for this development seems pretty obvious, namely, the study of whole numbers in modern mathematics is inextricably bound up with the theory of functions of a complex variable, which theory we have already seen has a good chance of forming the basis of the physics of the future. The working out of this idea would lead to a connection between atomic theory and cosmology."

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

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

A. LeClair, "Riemann Hypothesis and random walks: The zeta case" (preprint 01/2016)

[abstract:] "In previous work it was shown that if certain series based on sums over primes of non-principal Dirichlet characters have a conjectured random walk behavior, then the Euler product formula for its $L$-function is valid to the right of the critical line $\Re (s) > 1/2$, and the Riemann Hypothesis for this class of $L$-functions follows. Building on this work, here we propose how to extend this line of reasoning to the Riemann zeta function and other principal Dirichlet $L$-functions. We use our results to argue that $S_\delta (t) \equiv \lim_{\delta \to 0^+} \dfrac{1}{\pi} \arg\zeta(\tfrac{1}{2}+ \delta + it) = O(1)$, and that it is nearly always on the principal branch. We conjecture that a 1-point correlation function of the Riemann zeros has a normal distribution. This leads to the construction of a probabilistic model for the zeros. Based on these results we describe a new algorithm for computing very high Riemann zeros as a kind of stochastic process, and we calculate the $10^{100}$-th zero to over 100 digits."

F. Grosshans, T. Lawson, F. Morain and B. Smith, "Factoring safe semiprimes with a single quantum query" (preprint 11/2015)

[abstract:] "Shor's factoring algorithm (SFA), by its ability to efficiently factor large numbers, has the potential to undermine contemporary encryption. At its heart is a process called order finding, which quantum mechanics lets us perform efficiently. SFA thus consists of a quantum order finding algorithm (QOFA), bookended by classical routines which, given the order, return the factors. But, with probability up to 1/2, these classical routines fail, and QOFA must be rerun. We modify these routines using elementary results in number theory, improving the likelihood that they return the factors. We present a new quantum factoring algorithm based on QOFA which is better than SFA at factoring safe semiprimes, an important class of numbers used in RSA encryption (and reputed to be the hardest to factor). With just one call to QOFA, our algorithm almost always factors safe semiprimes. As well as a speed-up, improving efficiency gives our algorithm other, practical advantages: unlike SFA, it does not need a randomly picked input, making it simpler to construct in the lab; and in the (unlikely) case of failure, the same circuit can be rerun, without modification. We consider generalising this result to other cases, although we do not find a simple extension, and conclude that SFA is still the best algorithm."

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

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

A. Abdesselam, "Towards three-dimensional conformal probability" (preprint 10/2015)

[abstract:] "In this outline of a program, based on rigorous renormalization group theory, we introduce new definitions which allow one to formulate precise mathematical conjectures related to conformal invariance as studied by physicists in the area known as higher-dimensional conformal bootstrap which has developed at a breathtaking pace over the last five years. We also explore a second theme, intimately tied to conformal invariance for random distributions, which can be construed as a search for a very general first and second-quantized Kolmogorov–Chentsov Theorem. First-quantized refers to regularity of sample paths. Second-quantized refers to regularity of generalized functionals or Hida distributions and relates to the operator product expansion. Finally, we present a summary of progress made on a $p$-adic hierarchical model and point out possible connections to number theory."

A. Jollivet and V. Sharafutdinov, "An inequality for the zeta function of a planar domain" (preprint 10/2015)

[abstract:] "We consider the zeta function $\zeta\_\Omega$ for the Dirichlet-to-Neumann operator of a simply connected planar domain $\Omega$ bounded by a smooth closed curve.We prove non-negativeness and growth properties for $\zeta\_\Omega(s)-2\big({L(\partial \Omega)\over 2\pi}\big)^s\zeta\_R(s)\ (s\leq-1)$, where $L(\partial \Omega)$ is the length of the boundary curve and $\zeta\_R$ stands for the classical Riemann zeta function. Two analogs of these results are also provided."

T. Okazaki, "AdS2/CFT1, Whittaker vector and Wheeler–De Witt equation" (preprint 10/2015)

[abstract:] "We study the energy representation of conformal quantum mechanics as the Whittaker vector without specifying classical Lagrangian. We show that a generating function of expectation values among two excited states of the dilatation operator in conformal quantum mechanics is a solution to the Wheeler–DeWitt equation and it corresponds to the AdS2 partition function evaluated as the minisuperspace wave function in Liouville field theory. We also show that the dilatation expectation values in conformal quantum mechanics lead to the asymptotic smoothed counting function of the Riemann zeros."

Dynamics, Geometry and Number Theory, 13–17 June 2016, Institut Henri Poincaré, Paris, France

R.A. Ribeiro Correia de Sousa, "Prime numbers: A particle in a box and the complex wave model", Journal of Mathematics Research 7 (4) (2015)

[abstract:] "Euler's formula establishes the relationship between the trigonometric function and the exponential function. In doing so unifies two waves, a real and an imaginary one, that propagate through the Complex number set, establishing relation between integer numbers. A complex wave, if anchored by zero and by a defined integer number $N$, only can assume certain oscillation modes. The first mode of oscillation corresponds always to a $N$ prime number and the other modes to its multiples.

\begin{center}
$$\psi (x)=x e^{i\left(\frac{n \pi }{N}x\right)}$$
\end{center}

Under the above described conditions, these waves and their admissible oscillation modes allows for primality testing of integer numbers, the deduction of a new formula $\pi(x)$ for counting prime numbers and the identification of patterns in the prime numbers distribution with computing time gains in the calculations. In this article, four theorems and one rule of factorizing are put forward with consequences for prime number signaling, counting and distribution. Furthermore, it is establish the relationship between this complex wave with a time independent semi-classical harmonic oscillator, in which the spectrum of the allowed energy levels are always only prime numbers. Thus, it is affirmative the reply to the question if the prime numbers distribution is related to the energy levels of a physical system."
[abstract:] "In this paper we have described some interesting mathematical connections with various expressions of some sectors of string theory and relativistic quantum gravity, principally with the Palumbo–Nardelli model applied to the bosonic strings and the superstrings, and some parts of the theory of the cubic equation. In Appendix A, we have described the mathematical connections with some equations concerning the possible relativistic theory of quantum gravity. In conclusion in Appendix B, we have described a proof of Fermat's Last Theorem for the cubic equation case $n = 3$."