statistical mechanics
and number theory

the Riemann zeta function interpreted as a partition function

lattice-related number theory (involving Ising models, percolation, etc.)

integer partition problems and physics

entropy and number theory

number theory and statistical mechanics – general

probabilistic number theory

the Riemann zeta function interpreted as a partition function

One of the earliest, and perhaps most significant, examples of number theory influencing the development of physics was the application of Pólya's work on the Riemann zeta function to the theory of phase transitions by Lee and Yang in the early 1950's.

In 1951-2, Lee and Yang were developing this theory, and Mark Kac became aware of their conjecture which was later to become the "Lee-Yang circle theorem". It brought to his mind a theorem from Pólya's paper "Bemerkung uber die integraldarstellung der Riemannschen zeta-funktion" ("Remarks on the integral representation of Riemann's zeta function"). He realised that a slight modification of Pólya's proof could be used to prove a special case of Lee and Yang's conjecture, and brought this to their attention.

Lee and Yang were then able to adapt the reasoning and, within a couple of weeks, produce a proof of their general theorem. Kac says, "I recall Professor Yang telling me at the time that Hilfsatz II of Pólya...was one essential ingredient in their proof.".

Lee-Yang theorems provide restrictions on the locations of zeros of the analytic continuation of (real) partition functions associated with systems in statistical mechanics. Such complex zeros are associated with phase transitions. The circle theorem mentioned above states that, for certain systems, all such zeros must lie on a unit circle, but there are other cases where the zeros are restricted to a line. Recall that the Riemann Hypothesis seeks to restrict the location of zeros of the Riemann zeta function to a line in the complex plane. This parallel has turned out to be quite significant, as we shall go on to see.

In equilibrium statistical mechanics, the fundamental object of study for a system is its partition function. In the theory of the distribution of primes, the fundamental object is the Riemann zeta function. In the following paper, Bernard Julia introduced an abstract "numerical gas", called the free Riemann gas, whose constituent particles are prime numbers, and whose partition function is identical to the zeta function. This is a remarkably simple and natural construction, and leads to a number of suggestive parallels between various aspects of statistical mechanics and analytic number theory.

B.L. Julia, "Statistical theory of numbers", from Number Theory and Physics, M. Waldschmidt, et. al. (eds.), Springer Proceedings in Physics 47 (Springer, 1989) [outline]

B.L. Julia, "Theories statistique et thermodynamique des nombres", in: Conference de Strasbourg en l'honneur de P. Cartier, Proc. IRMA-RCP25, Vol. 44 (1993). 

"We shall in fact bring a large chunk of theoretical physics technology to bear on important mathematical problems and conversely one hopes to learn from a century of analytic number theory to understand better several issues of modern physics like the quark-gluon plasma transition and the Hagedorn critical temperature."

B.L. Julia, "Thermodynamic limit in number theory: Riemann-Beurling gases", Physica A 203 425-436.

"We study the grand canonical version of a solved statistical model, the Riemann gas: a collection of bosonic oscillators with energies the logarithms of the prime numbers. The introduction of a chemical potential \mu amounts to multiply each prime by $e^{\mu}$, the corresponding gases could be called Beurling gases because they are defined by the choice of appropriate generalized primes when considered as canonical ensembles; one finds generalized Hagedorn singularities in the temperature. The discrete spectrum can be treated as continuous in its high energy region; this approximation allows us to study the high energy level density and is applied to Beurling gases. It is expected to be accurate for the high temperature behaviour. One model (the logarithmic gases) will be studied in more detail, it corresponds to the choice of all the integers strictly larger than one as Beurling primes; we give an explicit formula for its grand canonical thermodynamic potential $F - \mu N$ in terms of a hypergeometric function and check the approximation on the Hagedorn phenomenon. Related physical situations include string theories and quark deconfinement where one needs a better understanding of the nature of the Hagedorn transitions."

"Theorie analytique des nombres et mecanique statistique quantique" - a brief discussion (in French) of Julia's work and related issues.

Unknown to Julia at the time of his 1989 article, George Mackey had arrived at the same idea some years earlier in the following book. Julia points out that whereas Mackey treats only the bosonic case, his own work also considers the fermionic case. Mackey's book draws a number of additional parallels between statistical mechanics and number theory.

G.W. Mackey, Unitary Group Representation in Physics, Probability, and Number Theory (Benjamin, 1978). 

"...Our main point here is that one could have been led to the main outline of the proof of the prime number theorem by using the physical interpretation of Laplace transforms provided by statistical mechanics. In particular, the function -zeta'/zeta whose representation as a Dirichlet series (Laplace transform with discrete measure) plays a central role in the proof has a direct physical interpretation as the internal energy function." (p.300) 

Donald Spector, also unaware of Mackey's work, made a number of closely-related discoveries at the same time as Julia.

D. Spector, "Supersymmetry and the Möbius inversion function", Communications in Mathematical Physics 127 (1990) 239.

"We show that the Möbius inversion function of number theory can be interpreted as the operator (-1)F in quantum field theory...We will see in this paper that the function...has a very natural interpretation. In the proper context, it is equivalent to (-1)F, the operator that distinguishes fermionic from bosonic states and operators, with the fact that mu(n) = 0 when n is not squarefree being equivalent to the Pauli exclusion principle...One of the results we obtain is equivalent to the prime number theorem, one of the central achievements of number theory, in which the asymptotic density of prime numbers is computed."

M. Wolf, "Applications of statistical mechanics in prime number theory"

This is a summary of a recent preprint. It's a review article based on a lecture given in Budapest earlier this year (2000), and due to appear in Physica A. As well as covering the work of Julia, Spector, Knauf, etc. he introduces another abstract numerical gas. This is something like Julia's free Riemann gas, but instead of the energies of the particles (primes) being based on their magnitudes, they are based on the gaps between consecutive primes.

J. Baez's "This Weeks Finds in Mathematical Physics (Week 199)" discusses C* algebras and B. Julia's "free Riemann gas" concept, referencing my summary of the original article.

P. Leboeuf, A. G. Monastra and O. Bohigas, "The Riemannium", Regular and Chaotic Dynamics 6 (2001) 205-210.

[abstract:] "The properties of a fictitious, fermionic, many-body system based on the complex zeros of the Riemann zeta function are studied. The imaginary part of the zeros are interpreted as mean-field single-particle energies, and one fills them up to a Fermi energy EF. The distribution of the total energy is shown to be non-Gaussian, asymmetric, and independent of EF in the limit EF -> infinity. The moments of the limit distribution are computed analytically. The autocorrelation function, the finite energy corrections, and a comparison with random matrix theory are also discussed."

P. Leboeuf and A.G. Monastra, "Quantum thermodynamic fluctuations of a chaotic Fermi-gas model"

[abstract:] "We investigate the thermodynamics of a Fermi gas whose single-particle energy levels are given by the complex zeros of the Riemann zeta function. This is a model for a gas, and in particular for an atomic nucleus, with an underlying fully chaotic classical dynamics. The probability distributions of the quantum fluctuations of the grand potential and entropy of the gas are computed as a function of temperature and compared, with good agreement, with general predictions obtained from random matrix theory and periodic orbit theory (based on prime numbers). In each case the universal and non-universal regimes are identified."

G. Zhang, F. Martelli and S. Torquato, "Structure factor of the primes" (preprint 01/2018)

[abstract:] "Although the prime numbers are deterministic, they can be viewed, by some measures, as pseudo-random numbers. In this article, we numerically study the pair statistics of the primes using statistical-mechanical methods, especially the structure factor $S(k)$ in an interval $M=p=M+L$ with $M$ large, and $L/M$ smaller than unity. We show that the structure factor of the prime configurations in such intervals exhibits well-defined Bragg-like peaks along with a small "diffuse" contribution. This indicates that the primes are appreciably more correlated and ordered than previously thought. Our numerical results definitively suggest an explicit formula for the locations and heights of the peaks. This formula predicts infinitely many peaks in any non-zero interval, similar to the behavior of quasicrystals. However, primes differ from quasicrystals in that the ratio between the location of any two predicted peaks is rational. We also show numerically that the diffuse part decays slowly as $M$ or $L$ increases. This suggests that the diffuse part vanishes in an appropriate infinite-system-size limit."

S. Torquato, A. Scardicchio, C.E. Zachary, "Point processes in arbitrary dimension from fermionic gases, random matrix theory, and number theory" (preprint 09/2008)

[abstract:] "It is well known that one can map certain properties of random matrices, fermionic gases, and zeros of the Riemann zeta function to a unique point process on the real line. Here we analytically provide exact generalizations of such a point process in $d$-dimensional Euclidean space for any $d$, which are special cases of determinantal processes. In particular, we obtain the $n$-particle correlation functions for any n, which completely specify the point processes. We also demonstrate that spin-polarized fermionic systems have these same $n$-particle correlation functions in each dimension. The point processes for any $d$ are shown to be hyperuniform. The latter result implies that the pair correlation function tends to unity for large pair distances with a decay rate that is controlled by the power law $r^{-(d+1)}$. We graphically display one- and two-dimensional realizations of the point processes in order to vividly reveal their "repulsive" nature. Indeed, we show that the point processes can be characterized by an effective "hard-core" diameter that grows like the square root of $d$. The nearest-neighbor distribution functions for these point processes are also evaluated and rigorously bounded. Among other results, this analysis reveals that the probability of finding a large spherical cavity of radius $r$ in dimension $d$ behaves like a Poisson point process but in dimension $d+1$ for large r and finite $d$. We also show that as $d$ increases, the point process behaves effectively like a sphere packing with a coverage fraction of space that is no denser than $1/2^d$."

I. Bakas and M.J. Bowick, "Curiosities of arithmetic gases", Journal of Mathematical Physics 32 (7) (1991) 1881-1884

[abstract:] "Statistical mechanical systems with an exponential density of states are considered. The arithmetic analog of parafermions of arbitrary order is constructed and a formula for boson-parafermion equivalence is obtained using properties of the Riemann zeta function. Interactions (nontrivial mixing) among arithmetic gases using the concept of twisted convolutions are also introduced. Examples of exactly solvable models are discussed in detail."

A. Knauf, "Number theory, dynamical systems and statistical mechanics" (1998 lecture notes)

"In these lecture notes connections between the Riemann zeta function, motion in the modular domain and systems of statistical mechanics are presented." [extensive survey article]

work by Andreas Knauf, et. al. on number theoretical spin chains

J.-B. Bost and A. Connes, "Hecke Algebras, Type III factors and phase transitions with spontaneous symmetry breaking in number theory", Selecta Math. (New Series), 1 (1995) 411-457.

"In this paper, we construct a natural C*-dynamical system whose partition function is the Riemann zeta function. Our construction is general and associates to an inclusion of rings (under a suitable finiteness assumption) an inclusion of discrete groups (the associated ax + b groups) and the corresponding Hecke algebras of bi-invariant functions. The latter algebra is endowed with a canonical one parameter group of automorphisms measuring the lack of normality of the subgroup. The inclusion of rings Z provides the desired C*- dynamical system, which admits the zeta function as partition function and the Galois group Gal(Q cycl/ Q) of the cyclotomic extension Qcycl of Q as symmetry group. Moreover, it exhibits a phase transition with spontaneous symmetry breaking at inverse temperature beta = 1. The original motivation for these results comes from the work of B. Julia [J] (cf. also [Spe])."

D. Harari and E. Leichtnam "Extension du phenomene de brisure spontanee de symetrie de Bost-Connes au cas des corps global quelconques"

This generalises the result of Bost and Connes which interprets the Riemann zeta function as a partition function of a dynamical system (in the C*-algebra formalism) whereby the pole at s =1 is interpreted in terms of spontaneous symmetry breaking. The generalisation extends the result to general number fields, and is further improved in the following paper which generalises in such a way that the partition function becomes the appropriate Dedekind zeta function

M. Planat, P. Solé, S. Omar, "Riemann hypothesis and quantum mechanics" (preprint 12/2010)

[abstract:] "In their 1995 paper, Jean-Beno\^{i}t Bost and Alain Connes (BC) constructed a quantum dynamical system whose partition function is the Riemann zeta function $\zeta(\beta)$, where $\beta$ is an inverse temperature. We formulate Riemann hypothesis (RH) as a property of the low temperature Kubo-Martin-Schwinger (KMS) states of this theory. More precisely, the expectation value of the BC phase operator can be written as $$\phi_{\beta}(q)=N_{q-1}^{\beta-1} \psi_{\beta-1}(N_q), $$ where $N_q=\prod_{k=1}^qp_k$ is the primorial number of order $q$ and $ \psi_b $ a generalized Dedekind $\psi$ function depending on one real parameter $b$ as $$ \psi_b (q)=q \prod_{p \in \mathcal{P,}p | q}\frac{1-1/p^b}{1-1/p}.$$ Fix a large inverse temperature $\beta >2.$ The Riemann hypothesis is then shown to be equivalent to the inequality $$ \phi_\beta (N_q)\zeta(\beta-1) >e^\gamma \log \log N_q, $$ for $q$ large enough. Under RH, extra formulas for high temperatures KMS states ($1.5< \beta <2$) are derived."

P. Cohen "Dedekind zeta functions and quantum statistical mechanics"

P.B. Cohen, "A C*-dynamical system with Dedekind zeta partition function and spontaneous symmetry breaking", soumis aux Actes des Journees Arithmetiques de Limoges, 1997. Preprint de l'IRMA de l'UST de Lille.

A. Connes and M. Marcolli, "From Physics to Number Theory via Noncommutative Geometry. Part I: Quantum Statistical Mechanics of Q-lattices" (preprint 04/04)

[abstract:] "This is the first installment of a paper in three parts, where we use noncommutative geometry to study the space of commensurability classes of Q-lattices and we show that the arithmetic properties of KMS states in the corresponding quantum statistical mechanical system, the theory of modular Hecke algebras, and the spectral realization of zeros of L-functions are part of a unique general picture. In this first chapter we give a complete description of the multiple phase transitions and arithmetic spontaneous symmetry breaking in dimension two. The system at zero temperature settles onto a classical Shimura variety, which parameterizes the pure phases of the system. The noncommutative space has an arithmetic structure provided by a rational subalgebra closely related to the modular Hecke algebra. The action of the symmetry group involves the formalism of superselection sectors and the full noncommutative system at positive temperature. It acts on values of the ground states at the rational elements via the Galois group of the modular field."

M. Marcolli and A. Connes, "From physics to number theory via noncommutative geometry. Part II: Renormalization, the Riemann-Hilbert correspondence, and motivic Galois theory", from Frontiers in Number Theory, Physics, and Geometry: On Random Matrices, Zeta Functions, and Dynamical Systems (Springer, 2006)

E. Ha and F. Paugam, "Bost-Connes-Marcolli systems for Shimura varieties" (preprint 03/05)

[abstract:] "We construct a Quantum Statistical Mechanical system $(A,\sigma_t)$ analogous to the Bost-Connes-Marcolli the case of Shimura varieties. Along the way, we define a new Bost-Connes system for number fields which has the "correct" symmetries and "correct" partition function. We give a formalism that applies to general Shimura data (G,X). The object of this series of papers is to show that these systems have phase transitions and spontaneous symmetry breaking, and to classify their KMS states, at least for low temperature."   [additional background information]

J. Lagarias, "Number theory zeta functions and dynamical zeta functions", in Spectral Problems in Geometry and Arithmetic (T. Branson, ed.), Contemporary Math. 237 (AMS, 1999) 45-86

[abstract:] "We describe analogies between number theory zeta functions, dynamical zeta functions,and statistical mechanics zeta functions, with emphasis on multi-variable zeta functions. We mainly consider two-variable zeta functions $\zeta_{f}(z,s)$ in which the variable $z$ is a "geometric variable", while the variable $s$ is an "arithmetic variable". The $s$-variable has a thermodynamic interpretation, in which $s$ parametrizes a family of energy functions $\phi_{s}$. We survey results on the analytic continuation and location of zeros and poles of two-variable zeta functions for four examples connected with number theory. These examples are (1) the beta transformation $f(x) = \beta x$ (mod 1), (2) the Gauss continued fraction map $f(x) = 1/x$ (mod 1), (3) zeta functions of varieties over finite fields, and (4) Riemann zeta function."

Although the following does not deal with statistical mechanics as such, the author (seemingly unaware of the works of Mackey, Julia, and Spector) presents an exactly analogous interpretation of the Riemann zeta function as a partition function, in the context of quantum entanglement:

Daniel Fivel, "The prime factorization property of entangled quantum states"

"Completely entangled quantum states are shown to factorize into tensor product of entangled states whose dimensions are powers of prime numbers...We consider processes in which factors are exchanged between entangled states and study canonical ensembles in which these processes occur. It is shown that the Riemann zeta function is the appropriate partition function and that the Riemann hypothesis makes a prediction about the high temperature contribution of modes of large dimension."

J.J. Garcia Moreta, Chebyshev Partition function: A connection between statistical physics and Riemann Hypothesis" (preprint 09/2006)

[abstract:] "In this paper we present a method to obtain a possible self-adjoint Hamiltonian operator so its energies satisfy Z(1/2+iE_n)=0, which is an statement equivalent to Riemann Hypothesis, first we use the explicit formula for the Chebyshev function Psi(x) and apply the change x=exp(u), after that we consider an Statistical partition function involving the Chebyshev function and its derivative so Z=Tr(exp(-BH), from the integral definition of the partition function Z we try to obtain the Hamiltonian operator assuming that H=P^{2}+V(x) by proposing a Non-linear integral equation involving Z(B) and V(x)."

J.G. Dueñas and N. F. Svaiter, "Zeros of the partition function in the randomized Riemann gas" (preprint 02/2014)

[abstract:] "An arithmetic gas is a second quantized mechanical system where the partition function is a Dirichlet series of a given arithmetic function. One example of such a system is known as the bosonic Riemann gas. We assume that the hamiltonian of the bosonic Riemann gas has a random variable with some probability distribution over an ensemble of hamiltonians. We discuss the singularity structure for the average free energy density of this arithmetic gas in the complex $\beta$ plane. First, assuming the Riemann hypothesis, there is a clustering of singular points along the imaginary axis coming from the non-trivial zeros of the Riemann zeta function on the critical line. This singularity structure associated to the zeros of the partition functions of the ensemble in the complex $\beta$ plane are the Fisher zeros. Second, there are also logarithmic singularities due to the poles of the Riemann zeta functions associated to the ensemble of hamiltonians. Finally we present the average energy density of the system."

A.I. Solomon, G.E.H. Duchamp, P. Blasiak, A. Horzela, K.A. Penson, "Hopf algebra structure of a model quantum field theory" (Talk presented by first-named author at 26th International Colloquium on Group Theoretical Methods in Physics, New York, June 2006. See cs.OH/0609107 for follow-up talk delivered by second-named author.)

[abstract:] "Recent elegant work on the structure of Perturbative Quantum Field Theory (PQFT) has revealed an astonishing interplay between analysis (Riemann Zeta functions), topology (Knot theory), combinatorial graph theory (Feynman Diagrams) and algebra (Hopf structure). The difficulty inherent in the complexities of a fully-fledged field theory such as PQFT means that the essential beauty of the relationships between these areas can be somewhat obscured. Our intention is to display some, although not all, of these structures in the context of a simple zero-dimensional field theory; i.e. a quantum theory of non-commuting operators which do not depend on spacetime. The combinatorial properties of these boson creation and annihilation operators, which is our chosen example, may be described by graphs, analogous to the Feynman diagrams of PQFT, which we show possess a Hopf algebra structure. Our approach is based on the partition function for a boson gas. In a subsequent note in these Proceedings we sketch the relationship between the Hopf algebra of our simple model and that of the PQFT algebra."

P. Dutta and D. Ghoshal, "Phase operator on $L_2(\mathbb{Q}_p$) and the zeroes of Fisher and Riemann" (preprint 02/2021)

[abstract:] "The distribution of the non-trivial zeroes of the Riemann zeta function, according to the Riemann hypothesis, is tantalisingly similar to the zeroes of the partition functions (Fisher and Yang–Lee zeroes) of statistical mechanical models studied by physicists. The resolvent function of an operator akin to the phase operator, conjugate to the number operator in quantum mechanics, turns out to be important in this approach. The generalised Vladimirov derivative acting on the space $L_2(\mathbb{Q}_p)$ of complex valued locally constant functions on the $p$-adic field is rather similar to the number operator. We show that a `phase operator' conjugate to it can be constructed on a subspace $L_2(p^{-1}\mathbb{Z}_p)$ of $L_2(\mathbb{Q}_p)$. We discuss (at physicists' level of rigour) how to combine this for all primes to possibly relate to the zeroes of the Riemann zeta function. Finally, we extend these results to the family of Dirichlet $L$-functions, using our recent construction of Vladimirov derivative like pseudodifferential operators associated with the Dirichlet characters."

S. Dwivedi, V. Kumar Singh and A. Roy, "Semiclassical limit of topological Rényi entropy in $3d$ Chern–Simons theory" (preprint 07/2020)

[abstract:] "We study the multi-boundary entanglement structure of the state associated with the torus link complement $S^3\setminus T_{p,q}$ in the set-up of three-dimensional $SU(2)$_k Chern–Simons theory. The focal point of this work is the asymptotic behavior of the Rényi entropies, including the entanglement entropy, in the semiclassical limit of $k\to\infty$. We present a detailed analysis for several torus links and observe that the entropies converge to a finite value in the semiclassical limit. We further propose that the large $k$ limiting value of the Rényi entropy of torus links of type $T_{p,pn}$ is the sum of two parts: (i) the universal part which is independent of $n$, and (ii) the non-universal or the linking part which explicitly depends on the linking number $n$. Using the analytic techniques, we show that the universal part comprises of Riemann zeta functions and can be written in terms of the partition functions of two-dimensional topological Yang–Mills theory. More precisely, it is equal to the Rényi entropy of certain states prepared in topological $2d$ Yang–Mills theory with $SU(2)$ gauge group. Further, the universal parts appearing in the large $k$ limits of the entanglement entropy and the minimum Rényi entropy for torus links $T_{p,pn}$ can be interpreted in terms of the volume of the moduli space of flat connections on certain Riemann surfaces. We also analyze the Rényi entropies of $T_{p,pn}$ link in the double scaling limit of $k\to\infty$ and $n\to\infty$ and propose that the entropies converge in the double limit as well."



C. Newman, "Gaussian correlation inequalities for ferromagnets", Zeitschrift für Wahrscheinlichkeitstheorie und Verwandte Gebiete 33 (1975) 75-93.

D. Williams summarises here, saying Newman "explained that if it could be shown that a certain probability density function is 'ferromagnetic', then the Riemann Hypothesis would follow." He further notes that this density function "arises fairly naturally in the study of Brownian motion."

C. Newman, "The GHS inequality and the Riemann hypothesis", Constructive Approximation 7 No.3 (1991) 389-399

[abstract:] "Let V(t) be the even function on (-$\infty,\infty)$ which is related to the Riemann xi-function by $\Xi (x/2)=4\int\sp{\infty}\sb{-\infty}\exp (ixt-V(t))dt.$ In a proof of certain moment inequalities which are necessary for the validity of the Riemann hypothesis, it was previously shown that $V'(t)/t$ is increasing on (0,$\infty)$. We prove a stronger property which is related to the GHS inequality of statistical mechanics, namely that $V'$ is convex on $[0,\infty)$. The possible relevance of the convexity of $V'$ to the Riemann hypothesis is discussed."

C.A. Tracy, "The emerging role of number theory in exactly solvable models in lattice statistical mechanics", Physica D: Nonlinear Phenomena 25 (1987) 1-19

[abstract:] "We review the Baxter model, the hard hexagon model and their multistate generalizations from a point of view that stresses the connection with modular functions and additive number theory. It is shown, for example, that various physical quantities in the hard hexagon model are all expressible in terms of modular functions. The use of Rogers-Ramanujan type identities in solvable models is also reviewed."

Nan-Xian Chen, Mi Li and Shao-jun Liu, "Phonon dispersions and elastic constants of Ni3Al and Möbius inversion", Physics Letters A 195 (1994) 135-143

[abstract:] "The Möbius inversion formulation corresponding to inequally weighted summations for solving some three-dimensional lattice problems introduced by Chen et al. [Phys. Lett. A 184 (1994) 347] has been used for the first time to obtain the pair potentials for fcc and L12 structures. The derivation is exact for radial potentials not only between identical atoms, but also between distinct atoms. We have tested this formulation for Ni3Al using the empirical total energy function in the Rose model. The phonon dispersions and the elastic constants have been evaluated based on our calculated pair potentials and the results are in good agreement with the experimental data. This method shows a convenient route from electronic structure calculation or empirical formula of binding energy curve to mechanical properties of materials. It also indicates the potential application of the number theory method to condensed matter physics."

Nan-Xian Chen, "Modified Möbius inverse formula and its applications in physics", Phys. Rev. Lett. 64, 1193-1195 (1990)

[abstract:] "A new theorem of inverse formula is introduced for a kind of infinite series. Thus some new results for important inverse problems in physics are presented in this paper. These are the inverse problems for obtaining the phonon density of states, the inverse blackbody radiation problem for remote sensing, and the solution for inverse Ewald summation. Of more importance, it shows the possibility of the application of number theory to physical problems."

There appears to have been a second letter in the same volume on the same topic (p. 3203), and further discussion from other authors:

A.J. Pindor, "Comment on 'Modified Möbius inverse formula and its applications in physics'", Phys. Rev. Lett. 66 (1991) 957

Ninham, et. al. begin their their 1992 survey paper:

"This paper was stimulated by a brief note of Chen, which attracted some interest. Chen showed how to effect the asymptotic solution of several standard inverse problems in statistical physics by invoking the Möbius inversion formula, an apparently obscure result of algebraic number theory. The cornerstone of Chen's analysis is equivalent to the assertion that, under modest hypotheses on the functions $\alpha$ and $\beta$, if

$\alpha(x) = \Sum_{n=1}^{\infty} \beta(nx)$ for all $x > 0$


$\beta(x) = \Sum_{n=1}^{\infty}\mu(n)\alpha(nx)$ for all $x > 0$

[where $\mu(n)$ is the Möbius function]

To number theorists this key result in Chen is utterly trivial and well known, and Chen is utterly trivial and well known, and Chen subsequently noted that the rather circuitous original derivation of equation 2 can be replaced by appeal to [a theorem of Hardy and Wright], yet for physicists not familiar with analysis buried in classics like Titchmarsh or Hardy and Wright some new magical tools seem to have been invented. Indeed the Editor of Nature suggested (volume 344 (1990) that by so calling in the treasure-trove of the old world some new insights of classical analysis might become accessible."

B.W. Ninham and B.D. Hughes, "Möbius, Mellin, and mathematical physics", Physica A: Statistical and Theoretical Physics 186 (1992) 441-481

[abstract:] "We examine some results and techniques of analytic number theory which have application, or potential application, in mathematical physics. We consider inversion formulae for lattice sums, various transformations of infinite series and products, functional equations and scaling relations, with selected applications in electrostatics and statistical mechanics. In the analysis, the Mellin transform and the Riemann zeta function play a key role."

M. Bazant, notes on applications of Möbius inversion in physics

"After more than a century confined to pure mathematics, the Möbius Inversion Formula of number theory and combinatorics is now finding applications in science. . . The juxtaposition of venerable knowledge in mathematics with recent insights from physics leads naturally to (i) a generalized 'Möbius Series Inversion Formula' containing all the previous cases and (ii) methods to overcome various limitations of the old theory for physical applications."

M. Bazant, "Lattice inversion problems with applications in solid state physics"
(involves Möbius function and inversion, Dirichlet multiplication)

Y. Wei, G. Yan and Q. Z. Yao, "Dirichlet inversion and lattice inversion problem", Computers and Mathematics with Applications, 41 (2001) 641-645

[abstract:] "Another application of Dirichlet multiplication is considered in this note. We show that Dirichlet inversion in number theory plays an important role in lattice inversion problem. With the help of this concept, lattice inversion problem becomes straightforward."

K. Iguchi, "Generalized Wigner lattices as a Riemann solid: Fractals in Hurwitz zeta function" (submitted to Modern Physics Letters B)

"We study the ground state configuration and the excitation energy gaps in the strong coupling limit of the extended Hubbard model with a long-range interaction in one dimension. As proved by Hubbard and Pokrovsky and Uimin, the ground state configuration is quasiperiodic and as proved by Bak and Bruinsma, the excitation energy has a finite gap which forms a devil's stair as a function of the density of particles in the system. We show that the quasiperiodicity and the fractal nature of the excitation energy come from the nature of the long-range interaction that is related to the fractal nature of the Hurwitz Zeta function and the Riemann Zeta function."

S. Ishiwata, S. Matsutani and Y. Ônishi, "Localized state of hard core chain and cyclotomic polynomial: hard core limit of diatomic Toda lattice", Phys. Lett. A 231 (1997) 208-216

[abstract:] "We consider a one-dimensional classical hard core chain with different alternating masses m and M. For a certain mass ratio M/m, there exists a localized state which consists of three adjacent particles and propagates. Then its mass ratio is given by a polynomial with integer coefficients, which turns out to be the cyclotomic polynomial. We can derived the complete series of such mass ratios."

L. Campos Venuti, "The best quasi-free approximation: reconstructing the spectrum from ground state energies" (preprint 01/2011)

[abstract:] "The sequence of ground state energy density at finite size, e_{L}, provides much more information than usually believed. Having at disposal $e_L$ for short lattice sizes, we show how to re-construct an approximate quasi-particle dispersion for any interacting model. The accuracy of this method relies on the best possible quasi-free approximation to the model, consistent with the observed values of the energy $e_L$. We also provide a simple criterion to assess whether such a quasi-free approximation is valid. Perhaps most importantly, our method is able to assess whether the nature of the quasi-particles is fermionic or bosonic together with the effective boundary conditions of the model. The success and some limitations of this procedure are discussed on the hand of the spin-1/2 Heisenberg model with or without explicit dimerization and of a spin-1 chain with single ion anisotropy. A connection with the Riemann Hypothesis is also pointed out."

F. Vericat, "A lattice gas of prime numbers and the Riemann Hypothesis" (preprint 11/2012)

[abstract:] "In recent years there has been some interest in applying ideas and methods taken from Physics in order to approach several challenging mathematical problems, particularly the Riemann Hypothesis, perhaps motived by the apparent inaccessibility to their solution from a full rigorous mathematical point of view. Most of these kind of contributions are suggested by some quantum statistical physics problems or by questions originated in chaos theory. In this note, starting from a very simple model of one-dimensional lattice gas and using the concept of equilibrium states as being described by Gibbs measures, we link classical statistical mechanics to the Riemann Hypothesis."

L. Bétermin and P. Zhang, "Minimization of energy per particle among Bravais lattices in $R^2$: Lennard–Jones and Thomas–Fermi cases" (preprint 02/2014)

[abstract:] "We study the two dimensional Lennard–Jones energy per particle of lattices and we prove that the minimizer among Bravais lattices with sufficiently large density is triangular and that is not the case for sufficiently small density. We give other results about the global minimizer of this energy. Moreover we study the energy per particle stemming from Thomas–Fermi model in two dimensions and we prove that the minimizer among Bravais lattices with fixed density is triangular. We use a result of Montgomery from number theory about the minimization of theta functions in the plane."

S. Boukraa and J-M. Maillard, "Selected non-holonomic functions in lattice statistical mechanics and enumerative combinatorics" (preprint 10/2015)

[abstract:] "We recall that the full susceptibility series of the Ising model, modulo powers of the prime 2, reduce to algebraic functions. We also recall the non-linear polynomial differential equation obtained by Tutte for the generating function of the $q$-coloured rooted triangulations by vertices, which is known to have algebraic solutions for all the numbers of the form $2+2\cos(j\pi;/n)$, the status of the $q = 4$ being unclear. We focus on the analysis of the $q = 4$ case, showing that the corresponding series is quite certainly non-holonomic. Along the line of a previous work on the susceptibility of the Ising model, we consider this $q = 4$ series modulo the first eight primes 2, 3,...,19, and show that this (probably non-holonomic) function reduces, modulo these primes, to algebraic functions. We conjecture that this probably non-holonomic function reduces to algebraic functions modulo (almost) every prime, or power of prime numbers. This raises the question to see whether such remarkable non-holonomic functions can be seen as ratio of diagonals of rational functions, or even rational, or algebraic, functions of diagonals of rational functions."

I. Vardi, "Deterministic percolation", Communications in Mathematical Physics 207 (1999) 43-66

[excerpt from introduction:] "...percolation theory has been of great interest in physics, as it is one of the simplest models to exhibit phase transitions. In this paper, I will examine how questions of percolation theory can be posed in a deterministic setting. Thus deterministic percolation is the study of unbounded walks on a single subset of a graph, e.g., defined by number theoretic conditions. This might be of interest in physics and probability theory as it studies percolation in a deterministic setting and in number theory where it can be interpreted as studying the disorder inherent in the natural numbers."

I. Vardi, "Prime percolation", Experimental Mathematics 7 (1998) 275-288

[abstract:] "This paper examines the question of whether there is an unbounded walk of bounded step size along Gaussian primes. Percolation theory predicts that for a low enough density of random Gaussian integers no walk exists, which suggests that no such walk exists along prime numbers, since they have arbitrarily small density over large enough regions. In analogy with the Cramer conjecture, I construct a random model of Gaussian primes and show that an unbounded walk of step size $k\sqrt{\log|z|}$ at $z$ exists with probability 1 if $k \gt \sqrt{2\pi\lambda_{c}}$ and does not exist with probability 1 if $k \lt \sqrt{2\pi\lambda_{c}}$ where $\lambda_{c}$ ~ 0.35 is a constant in continuum percolation, and so conjecture that the critical step size for Gaussian primes is also $\sqrt{2\pi\lambda_{c}\log|z|}$.

H.N.V. Temperley, "Further results on self-avoiding walks", Physica A: Statistical and Theoretical Physics 206 (1994) 350-358

[abstract:] "A Gaussian model of self-avoiding walks is studied. Not only is any cluster integral exactly evaluable, but whole sub-series can be evaluated exactly in terms of associated Riemann zeta functions. The results are compared with information recently obtained on self-avoiding walks on the plane square and simple cubic lattices and, as expected, are very similar. Use is made of the author's recent result that the reciprocal of the walks generating function is the generating function for irreducible cluster-sums. This is split into sub-series all of which have the same radius of convergence, and the significance of this is discussed."

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."

notes on the analogy between the functional equation of the Riemann zeta function and the Kramers-Wannier duality of statistical mechanics

J.-F. Burnol speculates on possible relationships between zeta and the Kramers-Wannier duality, Ising models etc.


statistical mechanics – general

D. Ruelle, "Zeta functions and statistical mechanics", Asterisque 40 (1976), 167-176.

D. Ruelle, "Is our mathematics natural? The case of equilibrium statistical mechanics", Bulletin of the AMS 19 (1988) 259-268.

excerpts from E. Schrödinger's Statistical Thermodynamics wherein appear special values of the Riemann zeta function

Y.V. Fyodorov, G.A. Hiary and J.P. Keating, "Freezing transition, characteristic polynomials of random matrices, and the Riemann zeta-function" (preprint 02/2012)

[abstract:] "We argue that the freezing transition scenario, previously explored in the statistical mechanics of $1/f$-noise random energy models, also determines the value distribution of the maximum of the modulus of the characteristic polynomials of large $N \times N$ random unitary (CUE) matrices. We postulate that our results extend to the extreme values taken by the Riemann zeta-function $\zeta(s)$ over sections of the critical line $s=1/2+it$ of constant length and present the results of numerical computations in support. Our main purpose is to draw attention to possible connections between the statistical mechanics of random energy landscapes, random matrix theory, and the theory of the Riemann zeta function."

B. Luque, I.G. Torre and L. Lacasa, "Phase transitions in number theory: From the birthday problem to Sidon sets" (preprint 10/2013)

[abstract:] "In this work, we show how number theoretical problems can be fruitfully approached with the tools of statistical physics. We focus on g-Sidon sets, which describe sequences of integers whose pairwise sums are different, and propose a random decision problem which addresses the probability of a random set of $k$ integers to be g-Sidon. First, we provide numerical evidence showing that there is a crossover between satisfiable and unsatisfiable phases which converts to an abrupt phase transition in a properly defined thermodynamic limit. Initially assuming independence, we then develop a mean field theory for the g-Sidon decision problem. We further improve the mean field theory, which is only qualitatively correct, by incorporating deviations from independence, yielding results in good quantitative agreement with the numerics for both finite systems and in the thermodynamic limit. Connections between the generalized birthday problem in probability theory, the number theory of Sidon sets and the properties of q-Potts models in condensed matter physics are briefly discussed."

C.E. Zachary, S. Torquato, "Hyperuniformity in point patterns and two-phase random heterogeneous media" (preprint 10/2009)

[abstract:] "Hyperuniform point patterns are characterized by vanishing infinite wavelength density fluctuations and encompass all crystal structures, certain quasi-periodic systems, and special disordered point patterns. This article generalizes the notion of hyperuniformity to include also two-phase random heterogeneous media. Hyperuniform random media do not possess infinite-wavelength volume fraction fluctuations, implying that the variance in the local volume fraction in an observation window decays faster than the reciprocal window volume as the window size increases. For microstructures of impenetrable and penetrable spheres, we derive an upper bound on the asymptotic coefficient governing local volume fraction fluctuations in terms of the corresponding quantity describing the variance in the local number density (i.e., number variance). Extensive calculations of the asymptotic number variance coefficients are performed for a number of disordered (e.g., quasiperiodic tilings, classical stealth disordered ground states, and certain determinantal point processes), quasicrystal, and ordered (e.g., Bravais and non-Bravais periodic systems) hyperuniform structures in various Euclidean space dimensions, and our results are consistent with a quantitative order metric characterizing the degree of hyperuniformity. We also present corresponding estimates for the asymptotic local volume fraction coefficients for several lattice families. Our results have interesting implications for a certain problem in number theory."

A. Le Méhauté, A. El Kaabouchi, L. Nivanen and Qiuping A. Wang, "Fractional dynamics, tiling equilibrium states and Riemann's zeta function" (preprint 07/09)

[abstract:] "It is argued that the generalisation of the mechanical principles to other variables than localisation, velocity and momentum leads to the laws of generalized dynamics under the condition of continuous and derivable space time. However, when the fractality arises, the mechanics principles may no more be extended especially because the time and space singularity appears on the boundary and creates curvature. There is no more equilibrium state, but only a horizon which might play a same role as equilibrium but does not close the problem - especially the problem of the invariance of the energy - which requires two complementary factors: a first one related to the closure in the dimensional space, and the second to scan dissymmetry stemming from the default of tiling the space time. A new discrete time arises from fractality. It leads irreversible thermodynamic properties. Space and time singularities lead to the relation between the above mentioned problematic and the Riemann zeta functions as well as its zeros."

A. LeClair, "Interacting Bose and Fermi gases in low dimensions and the Riemann hypothesis" (preprint, 11/2006)

[abstract:] "We apply the S-matrix based finite temperature formalism we recently developed to non-relativistic Bose and Fermi gases in 1+1 and 2+1 dimensions. In the 2+1 dimensional case, the free energy is given in terms of Roger's dilogarithm in a way analagous to the relativistic 1+1 dimensional case. The 1-d fermionic case with a quasi-periodic 2-body potential provides a physical framework for understanding the Riemann hypothesis."

A. Roussou, J. Smyrnakis, M. Magiropoulos, Nikolaos K. Efremidis, G.M. Kavoulakis, P. Sandin, M. Ögren and M. Gulliksson, "Excitation spectrum of a mixture of two Bose gases confined in a ring potential with interaction asymmetry" (preprint 11/2017)

[abstract:] "We study the rotational properties of a two-component Bose–Einstein condensed gas of distinguishable atoms which are confined in a ring potential using both the mean-field approximation, as well as the method of diagonalization of the many-body Hamiltonian. We demonstrate that the angular momentum may be given to the system either via single-particle, or "collective" excitation. Furthermore, despite the complexity of this problem, under rather typical conditions the dispersion relation takes a remarkably simple and regular form. Finally, we argue that under certain conditions the dispersion relation is determined via collective excitation. The corresponding many-body state, which, in addition to the interaction energy minimizes also the kinetic energy, is dictated by elementary number theory."

A.S. Mischenko, "Maslov's concept of phase transition from Bose–Einstein to Fermi–Dirac distribution: Results of interdisciplinary workshop in MSU" (preprint 11/2018)

[abstract:] "At the end of 2017, an interdisciplinary scientific seminar was organized at Moscow University, devoted to the study and development of a new scientific concept created by V.P. Maslov, allowing you to take a fresh look at the statistics of Bose–Einstein and Fermi–Dirac ideal gases. This new point of view allows us to interpret the indicated statistics as particular cases of statistical properties in number theory, on the one hand, and to indicate the limits of phase transitions from Bose to Fermi distributions."

D. Momeni, "Bose–Einstein condensation for an exponential density of states function and Lerch zeta function" (preprint 02/2019)

[abstract:] "I showed that how Bose-Einstein condensation (BEC) in a non interacting bosonic system with exponential density of the states function yields to a new class of Lerch zeta functions. By looking on the critical temperature, I suggeted a possible strategy to prove the ``Riemann hypothesis'' problem. In a theorem and a lemma I suggested that the classical limit $\hbar\to 0$Â of BEC can be used as a tool to find zeros of real part of the Riemann zeta function with complex argument. It reduces the Riemann hypothesis to a softer form. Furthermore I proposed a pair of creation-annihilation operators for BEC phenomena. These set of creation-annihilation operators is defined on a complex Hilbert space. They build a set up to interpret this type of BEC as a creation-annihilation phenomena of the virtual hypothetical particle."

M.J. Bowick, "Finite temperature strings"

[abstract:] "These are lecture notes for the 1992 Erice Workshop on String Quantum Gravity and Physics at the Planck Energy Scale. In this talk a review of earlier work on finite temperature strings was presented. Several topics were covered, including the canonical and microcanonical ensemble of strings, the behavior of strings near the Hagedorn temperature as well as speculations on the possible phases of high temperature strings. The connection of the string ensemble and, more generally, statistical systems with an exponentially growing density of states with number theory was also discussed."

M.N. Tran, M.V.N. Murthy, R.K. Bhaduri, "On the quantum density of states and partitioning an integer" (preprint 09/03)

[abstract:] "This paper exploits the connection between the quantum many-particle density of states and the partitioning of an integer in number theory. For N bosons in a one dimensional harmonic oscillator potential, it is well known that the asymptotic (N -> infinity) density of states is identical to the Hardy-Ramanujan formula for the partitions p(n), of a number n into a sum of integers. We show that the same statistical mechanics technique for the density of states of bosons in a power-law spectrum yields the partitioning formula for ps(n), the latter being the number of partitions of n into a sum of s-th powers of a set of integers. By making an appropriate modification of the statistical technique, we are also able to obtain ds(n) for distinct partitions. We find that the distinct square partitions d2(n) show pronounced oscillations as a function of n about the smooth curve derived by us. The origin of these oscillations from the quantum point of view is discussed. After deriving the Erdös-Lehner formula for restricted partitions for the s = 1 case by our method, we generalize it to obtain a new formula for distinct restricted partitions."

C. Weiss, S. Page, and M. Holthaus, "Factorising numbers with a Bose Einstein condensate" (preprint 03/04)

[abstract:] "The problem to express a natural number N as a product of natural numbers without regard to order corresponds to a thermally isolated non-interacting Bose gas in a one-dimensional potential with logarithmic energy eigenvalues. This correspondence is used for characterising the probability distribution which governs the number of factors in a randomly selected factorisation of an asymptotically large N. Asymptotic upper bounds on both the skewness and the excess of this distribution, and on the total number of factorisations, are conjectured. The asymptotic formulas are checked against exact numerical data obtained with the help of recursion relations. It is also demonstrated that for large numbers which are the product of different primes the probability distribution approaches a Gaussian, while identical prime factors give rise to non-Gaussian statistics."

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

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

B. Abdesselam and A. Chakrabarti, "A nested sequence of projectors (2): Multiparameter multistate statistical models, Hamiltonians, S-matrices" (preprint 01/06)

[abstract:] "Our starting point is a class of braid matrices, presented in a previous paper, constructed on a basis of a nested sequence of projectors. Statistical models associated to such N2 x N2 matrices for odd N are studied here. Presence of (N+3)(N-1)/2 free parameters is the crucial feature of our models, setting them apart from other well-known ones. There are N possible states at each site. The trace of the transfer matrix is shown to depend on (N-1)/2 parameters. For order r, N eigenvalues consitute the trace and the remaining Nr -N eigenvalues involving the full range of parameters come in zero-sum multiplets formed by the r-th roots of unity, or lower dimensional multiplets corresponding to factors of the order r when r is not a prime number. The modulus of any eigenvalue is of the form e\mu\theta, where \mu is a linear combination of the free parameters, \theta being the spectral parameter. For r a prime number an amusing relation of the number of multiplets with a theorem of Fermat is pointed out. Chain Hamiltonians and potentials corresponding to factorizable S matrices are constructed starting from our braid matrices. Perspectives are discussed."

E. Canessa, "Theory of analogous force on number sets" (preprint 07/03)

[abstract:] "A general statistical thermodynamic theory that considers given sequences of [natural numbers] to play the role of particles of known type in an isolated elastic system is proposed. By also considering some explicit discrete probability distributions px for natural numbers, we claim that they lead to a better understanding of probabilistic laws associated with number theory. Sequences of numbers are treated as the size measure of finite sets. By considering px to describe complex phenomena, the theory leads to derive a distinct analogous force fx on number sets proportional to $(\fract{\partial p_{x}}{\partial x})_{T}$ at an analogous system temperature T. In particular, this yields to an understanding of the uneven distribution of integers of random sets in terms of analogous scale invariance and a screened inverse square force acting on the significant digits. The theory also allows to establish recursion relations to predict sequences of Fibonacci numbers and to give an answer to the interesting theoretical question of the appearance of the Benford's law in Fibonacci numbers. A possible relevance to prime numbers is also analyzed."

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."

G.G. Szpiro, "The gaps between the gaps: some patterns in the prime number sequence", Physica A: Statistical Mechanics and its Applications 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"."

S. Ares and M. Castro, "Hidden structure in the randomness of the prime number sequence?", Physica A 360 (2006) 285

[abstract:] "We report a rigorous theory to show the origin of the unexpected periodic behavior seen in the consecutive differences between prime numbers. We also check numerically our findings to ensure that they hold for finite sequences of primes, that would eventually appear in applications. Finally, our theory allows us to link with three different but important topics: the Hardy-Littlewood conjecture, the statistical mechanics of spin systems, and the celebrated Sierpinski fractal."

K. Iguchi "Generalized Sommerfeld theory: Specific heat of a degenerate g-on gas in any dimension and the generalized Riemann zeta function", International Journal of Modern Physics B11, 3551-3580 (1997).

H.P. Baltes, E.R. Hilf and M. Pabst, "The long-time behaviour of the electric-field autocorrelation function in a finite photon gas", Applied Physics B: Lasers and Optics 3 (1974) 1432-0649

[note:] This involves the use of generalized Riemann zeta functions.

S. Nechaev and O. Vasilyev, "On metric structure of ultrametric spaces", J. Phys. A 37 (2004) 3783-3803

[abstract:] "In our work we have reconsidered the old problem of diffusion at the boundary of ultrametric tree from a "number theoretic" point of view. Namely, we use the modular functions (in particular, the Dedekind eta-function) to construct the "continuous" analog of the Cayley tree isometrically embedded in the Poincaré upper half-plane. Later we work with this continuous Cayley tree as with a standard function of a complex variable. In the frameworks of our approach the results of Ogielsky and Stein on dynamics on ultrametric spaces are reproduced semi-analytically/semi-numerically. The speculation on the new "geometrical" interpretation of replica n -> 0 limit is proposed."

P.J. Forrester, "Diffusion processes and the asymptotic bulk gap probability for the real Ginibre ensemble" (preprint 06/2013)

[abstract:] "It is known that the bulk scaling limit of the real eigenvalues for the real Ginibre ensemble is equal in distribution to the rescaled $t \to \infty$ limit of the annihilation process $A + A \to \emptyset$. Furthermore, deleting each particle at random in the rescaled $t \to \infty$ limit of the coalescence process $A + A \to A$, a process equal in distribution to the annihilation process results. We use these inter-relationships to deduce from the existing literature the asymptotic small and large distance form of the gap probability for the real Ginibre ensemble. In particular, the leading form of the latter is shown to be equal to $\exp(-(\zeta(3/2)/(2 \sqrt{2 \pi}))s)$, where $s$ denotes the gap size and $\zeta(z)$ denotes the Riemann zeta function. A determinant formula is derived for the gap probability in the finite $N$ case, and this is used to illustrate the asymptotic formulas against numerical computations."

H. Ono and H. Kuratsuji, "Statistical theory of 2-dimensional quantum vortex gas: non-canonical effect and generalized zeta function"

"The purpose of this paper is to present a quantum statistical theory of 2-dimensional vortex gas based on the generalized Hamiltonian dynamics recently developed...A remarkable consequence is that the partition function and related quantities are given in terms of the generalized Riemann zeta function. The topological phase transition is naturally understood as the pole structure of the zeta function."

B. Eckhardt, "Eigenvalue statistics in quantum ideal gases"

"The eigenvalue statistics of quantum ideal gases with single particle energies $e_n=n^\alpha$ are studied. A recursion relation for the partition function allows to calculate the mean density of states from the asymptotic expansion for the single particle density. For integer $\alpha>1$ one expects and finds number theoretic degeneracies and deviations from the Poissonian spacing distribution. By semiclassical arguments, the length spectrum of the classical system is shown to be related to sums of integers to the power $\alpha/(\alpha-1)$. In particular, for $\alpha=3/2$, the periodic orbits are related to sums of cubes, for which one again expects number theoretic degeneracies, with consequences for the two point correlation function."

M.V. Berry and P. Shukla, "Tuck's incompressibility function: statistics for zeta zeros and eigenvalues" (preprint 07/2008)

[abstract:] "For any function that is real for real $x$, positivity of Tuck's function $Q(x)=D'^2(x)/(D'^2(x)-D"(x) D(x))$ is a condition for the absence of the complex zeros close to the real axis. Study of the probability distribution $P(Q)$, for $D(x)$ with $N$ zeros corresponding to eigenvalues of the Gaussian unitary ensemble (GUE), supports Tuck's observation that large values of $Q$ are very rare for the Riemann zeros. $P(Q)$ has singularities at $Q=0$, $Q=1$ and $Q=N$. The moments (averages of $Q^m$) are much smaller for the GUE than for uncorrelated random (Poisson-distributed) zeros. For the Poisson case, the large-$N$ limit of $P(Q)$ can be expressed as an integral with infinitely many poles, whose accumulation, requiring regularization with the Lerch transcendent, generates the singularity at $Q=1$, while the large-$Q$ decay is determined by the pole closest to the origin. Determining the large-$N$ limit of $P(Q)$ for the GUE seems difficult."

A.L. Kholodenko, "Statistical mechanics of 2+1 gravity from Riemann zeta function and Alexander polynomial: Exact results"

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

M. Takahashi, "Correlation function and simplified TBA equations for XXZ chain" (preprint 12/2010)

[abstract:] "The calculation of the correlation functions of Bethe ansatz solvable models is very difficult problem. Among these solvable models spin 1/2 XXX chain has been investigated for a long time. Even for this model only the nearest neighbor and the second neighbor correlations were known. In 1990's multiple integral formula for the general correlations is derived. But the integration of this formula is also very difficult problem. Recently these integrals are decomposed to products of one dimensional integrals and at zero temperature, zero magnetic field and isotropic case, correlation functions are expressed by ${\rm log} 2$ and Riemann's zeta functions with odd integer argument $\zeta(3), \zeta(5), \zeta(7),....$. We can calculate density sub-matrix of successive seven sites. Entanglement entropy of seven sites is calculated. These methods can be extended to XXZ chain up to $n=4$. Correlation functions are expressed by the generalized zeta functions.

Several years ago I derived new thermodynamic Bethe ansatz equation for XXZ chain. This is quite different with Yang-Yang type TBA equations and contains only one unknown function. This equation is very useful to get the high temperature expansion. In this paper we get the analytic solution of this equation at $\Delta=0$."

C. Pineda and T. Prosen, "Non-universal level statistics in a chaotic quantum spin chain", Phys. Rev. E 76 (2007) 061127

[abstract:] "We study the level statistics of an interacting multi-qubit system, namely the kicked Ising spin chain, in the regime of quantum chaos. Long range quasi-energy level statistics show effects analogous to the ones observed in semi-classical systems due to the presence of classical periodic orbits, while short range level statistics display perfect statistical agreement with random matrix theory. Even though our system possesses no classical limit, our result suggest existence of an important non-universal system specific behavior at short time scale, which clearly goes beyond finite size effects in random matrix theory."

[author comment:] "It [attempts] to calculate the dimension of a Hilbert space associated with rotationally invariant systems of $n$ spins. The dimension was given in terms of the Möbius function."

A. Klümper, D. Nawrath and J. Suzuki, "Correlation functions of the integrable isotropic spin-1 chain: algebraic expressions for arbitrary temperature" (preprint 04/2013)

[abstract:] "We derive algebraic formulas for the density matrices of finite segments of the integrable $su(2)$ isotropic spin-1 chain in the thermodynamic limit. We give explicit results for the 2 and 3 site cases for arbitrary temperature $T$ and zero field. In the zero temperature limit the correlation functions are given in elementary form in terms of Riemann's zeta function at even integer arguments."

A.P.C. Malbouisson and J.M.C. Malbouisson, "Boundary dependence of the coupling constant and the mass in the vector N-component $(\lambda \phi^{4})_{D}$ theory", Journal of Physics A 35 (2002) 2263-2273.

[Abstract:] "Using the Matsubara formalism, we consider the massive $(\lambda \phi^{4})_{D}$ vector N component model in the large N limit, the system being confined between two infinite parallel planes. We investigate the behavior of the coupling constant as a function of the separation L between the planes. For the Wick-ordered model in D = 3 we are able to give an exact formula to the L-dependence of the coupling constant. For the non-Wick-ordered model we indicate how expressions for the coupling constant and the mass can be obtained for arbitrary dimension D in the small-L regime. Closed exact formulas for the L-dependent renormalized coupling constant and mass are obtained in D = 3 and their behaviors as functions of L are displayed. We are also able to obtainn in generic dimension D, an equation for the critical value of L corresponding to a second order phase transition in terms of the Riemann zeta-function. In D = 3 a renormalization is done and an explicit formula for the critical L is given."

S.A. Oprisal, "The classical gases in the Tsallis statistics using the generalized Riemann zeta functions", J. Phys. I France 7 (July 1997) 853-862.

[Abstract:] "In the last few years an increasing interest has been paid to fractal inspired statistics. Our aim is to describe some new insight obtained using Tsallis statistics. In the framework of the generalized statistics we described some properties of the Maxwell-Boltzmann gases. The behavior of the occupation numbers with respect to the temperature indicates similarities with Fermi gases. Using the Nernst theorem we also determine the fractal index of statistics."

P. Tempesta, "Group entropies, correlation laws and zeta functions" (preprint 05/2011)

[abstract:] "The notion of group entropy is proposed. It enables to unify and generalize many different definitions of entropy known in the literature, as those of Boltzmann–Gibbs, Tsallis, Abe and Kaniadakis. Other new entropic functionals are presented, related to nontrivial correlation laws characterizing universality classes of systems out of equilibrium, when the dynamics is weakly chaotic. The associated thermostatistics are discussed. The mathematical structure underlying our construction is that of formal group theory, which provides the general structure of the correlations among particles and dictates the associated entropic functionals. As an example of application, the role of group entropies in information theory is illustrated and generalizations of the Kullback–Leibler divergence are proposed. A new connection between statistical mechanics and zeta functions is established. In particular, Tsallis entropy is related to the classical Riemann zeta function."

S. Tanaka, "Distribution of the Riemann zeros represented by the Fermi gas" (preprint 10/2010)

[abstract:] "The multiparticle density matrices for degenerate, ideal Fermi gas system in any dimension are calculated. The results are expressed as a determinant form, in which a correlation kernel plays a vital role. Interestingly, the correlation structure of one-dimensional Fermi gas system is essentially equivalent to that observed for the eigenvalue distribution of random unitary matrices, and thus to that conjectured for the distribution of the non-trivial zeros of the Riemann zeta function. Implications of the present findings are discussed briefly. "

R. Pearson, "Number theory and critical exponents", Phys. Rev. B 22 (1980) 3465-3470

[abstract:] "The consequences of assuming p-adic analyticity for thermodynamic functions are discussed. Rules are given for determining the denominator of a rational critical exponent from the asymptotic behavior of the coefficients of series expansions. The example of the Hamiltonian Q-state Potts model is used to demonstrate the ideas of the paper."

P. Kleban, "Crossing probabilities in critical 2-D percolation and modular forms", Physica A 281 (2000) 242-251

[abstract:] "Crossing probabilities for critical 2-D percolation on large but finite lattices have been derived via boundary conformal field theory. These predictions agree very well with numerical results. However, their derivation is heuristic and there is evidence of additional symmetries in the problem. This contribution gives a preliminary examination some unusual modular behavior of these quantities. In particular, the derivatives of the "horizontal" and "horizontal-vertical" crossing probabilities transform as a vector modular form, one component of which is an ordinary modular form and the other the product of a modular form with the integral of a modular form. We include consideration of the interplay between conformal and modular invariance that arises."

P. Kleban and D. Zagier, "Crossing probabilities and modular forms" (preprint 09/02)

[abstract:] "We examine crossing probabilities and free energies for conformally invariant critical 2-D systems in rectangular geometries, derived via conformal field theory and Stochastic Löwner Evolution methods. These quantities are shown to exhibit interesting modular behavior, although the physical meaning of modular transformations in this context is not clear. We show that in many cases these functions are completely characterized by very simple transformation properties. In particular, Cardy's function for the percolation crossing probability (including the conformal dimension 1/3), follows from a simple modular argument. A new type of "higher-order modular form" arises and its properties are discussed briefly."

J. Hilgert, D. Mayer and H. Movasati, "Transfer operators for $\Gamma_0(n)$ and the Hecke operators for the period functions of $PSL(2,Z)$" (preprint, 03/03)

[abstract:] "In this article we report on a surprising relation between the transfer operators for the congruence subgroups $\Gamma_{0}(n)$ and the Hecke operators on the space of period functions for the modular group $\PSL(2,Z)$. For this we study special eigenfunctions of the transfer operators with eigenvalues +1, which are also solutions of the Lewis equations for the groups $\Gamma_{0}(n)$ and which are determined by eigenfunctions of the transfer operator for the modular group $\PSL(2,Z)$. In the language of the Atkin-Lehner theory of old and new forms one should hence call them old eigenfunctions or old solutions of Lewis equation. It turns out that the sum of the components of these old solutions for the group $\Gamma_{0}(n)$ determine for any n a solution of the Lewis equation for the modular group and hence also an eigenfunction of the transfer operator for this group."

D. Schumayer and D.A.W. Hutchinson, "Physics of the Riemann hypothesis", Rev. Mod. Phys. 83 (2011) 307–330

[abstract:] "Physicists become acquainted with special functions early in their studies. Consider our perennial model, the harmonic oscillator, for which we need Hermite functions, or the Laguerre functions in quantum mechanics. Here we choose a particular number theoretical function, the Riemann zeta function and examine its influence in the realm of physics and also how physics may be suggestive for the resolution of one of mathematics' most famous unconfirmed conjectures, the Riemann Hypothesis. Does physics hold an essential key to the solution for this more than hundred-year-old problem? In this work we examine numerous models from different branches of physics, from classical mechanics to statistical physics, where this function plays an integral role. We also see how this function is related to quantum chaos and how its pole-structure encodes when particles can undergo Bose–Einstein condensation at low temperature. Throughout these examinations we highlight how physics can perhaps shed light on the Riemann Hypothesis. Naturally, our aim could not be to be comprehensive, rather we focus on the major models and aim to give an informed starting point for the interested Reader."

R. M. Ziff, G. E. Uhlenbeck, and M. Kac, "The Bose–Einstein Gas, Revisited", Physics Reports 32C (1977)169-248

[This involves thermodynamic uses of the Riemann and Epstein zeta functions - see in particular section 3 starting with page 218.]

P. Cvitanovic, "Circle Maps: Irrationally Winding" from Number Theory and Physics, eds. C. Itzykson, et. al. (Springer, 1992)

See in particular sections 10.7 "Global Theory: Thermodynamic Averaging" and 10.12 "Farey Tree Thermodynamics"

[excerpt from 10.11, p.19] "The Farey series thermodynamics is of number theoretical interest, because the Farey series provide uniform coverings of the unit interval with rationals, and because they are closely related to the deepest problems in number theory, such as the Riemann hypothesis...

The Riemann hypothesis...would seem to have nothing to do with physicists' real mode-locking widths that we are interested in here. However, there is a real-line version of the Riemann hypothesis that lies very close to the mode-locking problem... The implications of this for the circle-map scaling theory have not been worked out, and it is not known whether some conjecture about the thermodynamics of irrational windings is equivalent to (or harder than) the Riemann hypothesis, but the danger lurks."

N. Chair, "Trigonometrical sums connected with one-dimensional lattice, the chiral Potts model and number theory using the residue operator" (preprint 06/2012)

[abstract:] "We have recently developed a method for obtaining exact two-point resistance on the square $N$-cycle, the complete graph minus $N$ edges of the opposite vertices, here, a similar method is used to obtain closed formulas of certain trigonometrical sums that arise in connection with one-dimensional lattice and the chiral Potts model. By modifying slightly, the previous computations, then, we are able to get new closed formulas for different trigonometrical sums, some of which appear in number theory."

B. Basu-Mallick, T. Bhattacharyya and D. Sen, "Clusters of bound particles in the derivative delta-function Bose gas" (preprint 10/2001)

[abstract:] "In this paper we discuss a novel procedure for constructing clusters of bound particles in the case of a quantum integrable derivative delta-function Bose gas in one dimension. It is shown that clusters of bound particles can be constructed for this Bose gas for some special values of the coupling constant, by taking the quasi-momenta associated with the corresponding Bethe state to be equidistant points on a single circle in the complex momentum plane. We also establish a connection between these special values of the coupling constant and some fractions belonging to the Farey sequences in number theory. This connection leads to a classification of the clusters of bound particles associated with the derivative delta-function Bose gas and allows us to study various properties of these clusters like their size and their stability under the variation of the coupling constant."

B. Basu-Mallick, T. Bhattacharyya and D. Sen, "Clusters of bound particles in a quantum integrable many-body system and number theory" (preprint 10/2014)

[abstract:] "We construct clusters of bound particles for a quantum integrable derivative delta-function Bose gas in one dimension. It is found that clusters of bound particles can be constructed for this Bose gas for some special values of the coupling constant, by taking the quasi-momenta associated with the corresponding Bethe state to be equidistant points on a single circle in the complex momentum plane. Interestingly, there exists a connection between the above mentioned special values of the coupling constant and some fractions belonging to the Farey sequences in number theory. This connection leads to a classification of the clusters of bound particles for the derivative delta-function Bose gas and the determination of various properties of these clusters like their size and their stability under a variation of the coupling constant."

A. Dias Ribeiro, "Level density of a Bose gas: Beyond the saddle point approximation" (preprint 11/2014)

[abstract:] "The present article is concerned with the use of approximations in the calculation of the many-body density of states (MBDS) of a system with total energy $E$, composed by $N$ bosons. In the mean-field framework, an integral expression for MBDS, which is proper to be performed by asymptotic expansions, can be derived. However, the standard second order steepest descent method cannot be applied to this integral when the ground-state is sufficiently populated. Alternatively, we derive a uniform formula for MBDS, which is potentially able to deal with this regime. In the case of the one-dimensional harmonic oscillator, using results found in the number theory literature, we show that the uniform formula improves the standard expression achieved by means of the second order method."

Y. Kong, "Packing dimers on $(2p+1)\times (2q+1)$ lattices" (preprint 10/2014)

[abstract:] "We use computational method to investigate the number of ways to pack dimers on odd-by-odd lattices. In this case, there is always a single vacancy in the lattices. We show that the dimer configuration numbers on $(2k+1)\times (2k+1)$ odd square lattices have some remarkable number-theoretical properties in parallel to those of close-packed dimers on $2k\times 2k$ even square lattices, for which exact solution exists. Furthermore, we demonstrate that there is an unambiguous logarithm term in the finite size correction of free energy of odd-by-odd lattice strips with any width $n = 1$. This logarithm term determines the distinct behavior of the free energy of odd square lattices. These findings reveal a deep and previously unexplored connection between statistical physics models and number theory, and indicate the possibility that the monomer-dimer problem might be solvable."

M.V.N. Murthy, M. Brack and R.K. Bhaduri, "On the asymptotic distinct prime partitions of integers" (preprint 04/2019)

[abstract:] "We discuss $Q(n)$, the number of ways a given integer $n$ may be written as a sum of distinct primes, and study its asymptotic form $Q_{as}(n)$ valid in the limit $n\to\infty$. We obtain $Q_{as}(n)$ by Laplace inverting the fermionic partition function of primes, in number theory called the generating function of the distinct prime partitions, in the saddle-point approximation. We find that our result of $Q_{as}(n)$, which includes two higher-order corrections to the leading term in its exponent and a pre-exponential correction factor, approximates the exact $Q(n)$ far better than its simple leading-order exponential form given so far in the literature."

R. Dong and M. Khalkhali, "Second quantization and the spectral action" (preprint 03/2019)

[abstract:] "We show that by incorporating chemical potentials one can extend the formalism of spectral action principle to bosonic second quantization. In fact we show that the von Neumann entropy, the average energy, and the negative free energy of the state defined by the bosonic, or fermionic, grand partition function can be expressed as spectral actions, and all spectral action coefficients can be given in terms of the modified Bessel functions. In the Fermionic case, we show that the spectral coefficients for the von Neumann entropy, in the limit when the chemical potential $\mu$ approaches to $0$, can be expressed in terms of the Riemann zeta function. This recovers a recent result of Chamseddine–Connes–van Suijlekom."

S. Ares and M. Castro, "Hidden structure in the randomness of the prime number sequence" (preprint 10/03)

[abstract:] "We report a rigorous theory to show the origin of the unexpected periodic behavior seen in the consecutive differences between prime numbers. We also check numerically our findings to ensure that they hold for finite sequences of primes, that would eventually appear in applications. Finally, our theory allows us to link with two different but important topics: the statistical mechanics of spin systems, and the celebrated Sierpinski fractal."

"The Prime Number Theorem obtained by statistical methods" - a heuristic argument from What is Mathematics? by Courant and Robbins

"By a procedure typical of...statistical mechanics we...[make] plausible of the distribution of primes."

number theory and entropy

archive      tutorial      mystery      new      search      home      contact