Abstract: We argue that the freezing transition scenario, previously conjectured to occur in the statistical mechanics of 1/f-noise random energy models, governs, after reinterpretation, the value distribution of the maximum of the modulus of the characteristic polynomials pN(θ) of large N×N random unitary (circular unitary ensemble) matrices UN; i.e. the extreme value statistics of pN(θ) when . In addition, we argue that it leads to multi-fractal-like behaviour in the total length μN(x) of the intervals in which |pN(θ)|>Nx,x>0, in the same limit. We speculate that our results extend to the large values taken by the Riemann zeta function ζ(s) over stretches of the critical line of given constant length and present the results of numerical computations of the large values of ). Our main purpose is to draw attention to the unexpected connections between these different extreme value problems.

Abstract: We study representation zeta functions of finitely generated, torsion-free nilpotent groups which are groups of rational points of unipotent group schemes over rings of integers of number fields. Using the Kirillov orbit method and $\frak{p}$-adic integration, we prove rationality and functional equations for almost all local factors of the Euler products of these zeta functions. We further give explicit formulae, in terms of Dedekind zeta functions, for the zeta functions of class-$2$-nilpotent groups obtained from three infinite families of group schemes, generalizing the integral Heisenberg group. As an immediate corollary, we obtain precise asymptotics for the representation growth of these groups, and key analytic properties of their zeta functions, such as meromorphic continuation. We express the local factors of these zeta functions in terms of generating functions for finite Weyl groups of type~$B$. This allows us to establish a formula for the joint distribution of three functions, or ``statistics'', on such Weyl groups. Finally, we compare our explicit formulae to $\frak{p}$-adic integrals associated to relative invariants of three infinite families of prehomogeneous vector spaces.

Abstract: The representation growth of a T -group is polynomial. We study the rate of polynomial growth and the spectrum of possible growth, showing that any rational number ? can be realized as the rate of polynomial growth of a class 2 nilpotent T -group. This is in stark contrast to the related subject of subgroup growth of T -groups where it has been shown that the set of possible growth rates is discrete in Q. We derive a formula for almost all of the local representation zeta functions of a T2-group with centre of Hirsch length 2. A consequence of this formula shows that the representation zeta function of such a group is finitely uniform. In contrast, we explicitly derive the representation zeta function of a specific T2-group with centre of Hirsch length 3 whose representation zeta function is not finitely uniform. We give formulae, in terms of Igusa's local zeta function, for the subring, left-, right- and two-sided ideal zeta function of a 2-dimensional ring. We use these formulae to compute a number of examples. In particular, we compute the subring zeta function of the ring of ?integers in a quadratic number field.

Abstract: The Stieltjes constants $\gamma_k(a)$ appear as the coefficients in the regular part of the Laurent expansion of the Hurwitz zeta function $\zeta(s,a)$ about $s=1$. We present series representations of these constants of interest to theoretical and computational analytic number theory. A particular result gives an addition formula for the Stieltjes constants. As a byproduct, expressions for derivatives of all orders of the Stieltjes coefficients are given. Many other results are obtained, including instances of an exponentially fast converging series representation for $\gamma_k=\gamma_k(1)$. Some extensions are briefly described, as well as the relevance to expansions of Dirichlet $L$ functions.

Abstract: This chapter provides an overview of the theory of hyperbolic zeta function of lattices. A functional equation for the hyperbolic zeta function of Cartesian lattice is obtained. Information about the history of the theory of the hyperbolic zeta function of lattices is provided. The relations with the hyperbolic zeta function of nets and Korobov optimal coefficients are considered.

Abstract: The present paper deals mainly with seven fundamental theorems of mathematical analysis, numerical analysis, and number theory, namely the generalized Parseval decomposition formula (GPDF), introduced 15 years ago, the well-known approximate sampling theorem (ASF), the new approximate reproducing kernel theorem, the basic Poisson summation formula, already known to Gaus, a newer version of the GPDF having a structure similar to that of the Poisson summation formula, namely, the Parseval decomposition–Poisson summation formula, the functional equation of Riemann’s zeta function, as well as the Euler–Maclaurin summation formula. It will in fact be shown that these seven theorems are all equivalent to one another, in the sense that each is a corollary of the others. Since these theorems can all be deduced from each other, one of them has to be proven independently in order to verify all. It is convenient to choose the ASF, introduced in 1963. The epilogue treats possible extensions to the more general contexts of reproducing kernel theory and of abstract harmonic analysis, using locally compact abelian groups. This paper is expository in the sense that it treats a number of mathematical theorems, their interconnections, their equivalence to one another. On the other hand, the proofs of the many intricate interconnections among these theorems are new in their essential steps and conclusions.

Abstract: The theory of “zeta functions of fractal strings” has been initiated by the first author in the early 1990s and developed jointly with his collaborators during almost two decades of intensive research in numerous articles and several monographs. In 2009, the same author introduced a new class of zeta functions, called “distance zeta functions,” which since then has enabled us to extend the existing theory of zeta functions of fractal strings and sprays to arbitrary bounded (fractal) sets in Euclidean spaces of any dimension. A natural and closely related tool for the study of distance zeta functions is the class of “tube zeta functions,” defined using the tube function of a fractal set. These three classes of zeta functions, under the name of “fractal zeta functions,” exhibit deep connections with Minkowski contents and upper box dimensions, as well as, more generally, with the complex dimensions of fractal sets. Further extensions include zeta functions of relative fractal drums, the box dimension of which can assume negative values, including minus infinity. We also survey some results concerning the existence of the meromorphic extensions of the spectral zeta functions of fractal drums, based in an essential way on earlier results of the first author on the spectral (or eigenvalue) asymptotics of fractal drums. It follows from these results that the associated spectral zeta function has a (nontrivial) meromorphic extension, and we use some of our results about fractal zeta functions to show the new fact according to which the upper bound obtained for the corresponding abscissa of meromorphic convergence is optimal.

Abstract: We shall prove an unconditional basic result related to the value- distributions of {(L′∕L)(s, χ)} χ and of {(ζ′∕ζ)(s + iτ)} τ , where χ runs over Dirichlet characters with prime conductors and τ runs over R. The result asserts that the expected density function common for these distributions are in fact the density function in an appropriate sense. Under the generalized Riemann hypothesis, stronger results have been proved in our previous articles, but our present result is unconditional.

Abstract: We construct a Hamiltonian H R whose discrete spectrum contains, in a certain limit, the Riemann zeros. H R is derived from the action of a massless Dirac fermion living in a domain of Rindler spacetime, in 1 + 1 dimensions, which has a boundary given by the world line of a uniformly accelerated observer. The action contains a sum of delta function potentials that can be viewed as partially reflecting moving mirrors. An appropriate choice of the accelerations of the mirrors, provide primitive periodic orbits that are associated with the prime numbers p, whose periods, as measured by the observerʼs clock, are . Acting on the chiral components of the fermion , H R becomes the Berry–Keating Hamiltonian , where x is identified with the Rindler spatial coordinate and with the conjugate momentum. The delta function potentials give the matching conditions of the fermion wave functions on both sides of the mirrors. There is also a phase shift for the reflection of the fermions at the boundary where the observer sits. The eigenvalue problem is solved by transfer matrix methods in the limit where the reflection amplitudes become infinitesimally small. We find that, for generic values of , the spectrum is a continuum where the Riemann zeros are missing, as in the adelic Connes model. However, for some values of , related to the phase of the zeta function, the Riemann zeros appear as discrete eigenvalues that are immersed in the continuum. We generalize this result to the zeros of Dirichlet L-functions, which are associated to primitive characters, that are encoded in the reflection coefficients of the mirrors. Finally, we show that the Hamiltonian associated to the Riemann zeros belongs to class AIII, or chiral GUE, of the Random Matrix Theory.

Abstract: The universality theoremasserts that the lower density of any set of shifts of the Riemann zeta-function which approximate a given analytic function with accuracy ɛ > 0 is strictly positive. It is proved that this set has strictly positive density for all but at most countably many ɛ > 0.

Abstract: On the assumption of the Riemann hypothesis, we generalize a central limit theorem of Fujii regarding the number of zeroes of Riemann's zeta function that lie in a mesoscopic interval. The result mirrors results of Spohn and Soshnikov and others in random matrix theory. In an appendix we put forward some general theorems regarding our knowledge of the zeta zeroes in the mesoscopic regime.

Abstract: We investigate the behavior of the Euler products of the Riemann zeta function and Dirichlet L-functions on the critical line. A refined version of the Riemann hypothe- sis, which is named "the Deep Riemann Hypothesis", is examined. We also study various analogs for global function fields. We give an interpretation for the nontrivial zeros from the viewpoint of statistical mechanics. Mathematics Subject Classification (2000). 11M06.

Abstract: We investigate the distribution of the logarithmic derivative of the Riemann zeta-function on the line Re(s)=\sigma, where \sigma, lies in a certain range near the critical line \sigma=1/2. For such \sigma, we show that the distribution of \zeta'/\zeta(s) converges to a two-dimensional Gaussian distribution in the complex plane. Upper bounds on the rate of convergence to the Gaussian distribution are also obtained.

Abstract: We establish pointwise and distributional fractal tube formulas for a large class of compact subsets of Euclidean spaces of arbitrary dimensions. These formulas are expressed as sums of residues of suitable meromorphic functions over the complex dimensions of the compact set under consideration (i.e., over the poles of its fractal zeta function). Our results generalize to higher dimensions (and in a significant way) the corresponding ones previously obtained for fractal strings by the first author and van Frankenhuijsen. They are illustrated by several examples and applied to yield a new Minkowski measurability criterion.

Abstract: It is known that ζ(1 + it) (log t) when t 1. This paper provides a new explicit estimate |ζ(1+it)| ≤ 34 log t, for t ≥ 3. This gives the best upper bound on |ζ(1 + it)| for t ≤ 102·105 .

Abstract: Chebyshev observed in a letter to Fuss that there tends to be more primes of the form 4n+3 than of the form 4n+1. The general phenomenon, which is referred to as Chebyshev’s bias, is that primes tend to be biased in their distribution among the different residue classes modq. It is known that this phenomenon has a strong relation with the low-lying zeros of the associated L-functions, that is, if these ‘L‘-functions have zeros close to the real line, then it will result in a lower bias. According to this principle one might believe that the most biased prime number race we will ever find is the Li(x) versus π(x) race, since the Riemann zeta function is the ‘L‘-function of rank one having the highest first zero. This race has density 0.99999973…, and we study the question of whether this is the highest possible density. We will show that it is not the case; in fact, there exist prime number races whose density can be arbitrarily close to 1. An example of a race whose density exceeds the above number is the race between quadratic residues and nonresidues modulo 4849845, for which the density is 0.999999928…. We also give fairly general criteria to decide whether a prime number race is highly biased or not. Our main result depends on the generalized Riemann hypothesis and a hypothesis on the multiplicity of the zeros of a certain Dedekind zeta function. We also derive more precise results under a linear independence hypothesis.

Abstract: For each prime p and each embeddingof the multiplicative group of an algebraic closure of Fp as complex roots of unity, we construct a p-adic indecomposable representation �� of the integral BC-system as additive endo- morphisms of the big Witt ring of ¯ Fp. The obtained representations are the p-adic analogues of the complex, extremal KMS1 states of the BC-system. The role of the Riemann zeta function, as partition function of the BC-system over C is replaced, in the p-adic case, by the p-adic L-functions and the poly- logarithms whose values at roots of unity encode the KMS states. We use Iwasawa theory to extend the KMS theory to a covering of the completion Cp of an algebraic closure of Qp. We show that our previous work on the hyper- ring structure of the adele class space, combines with p-adic analysis to refine the space of valuations on the cyclotomic extension of Q as a noncommuta- tive space intimately related to the integral BC-system and whose arithmetic geometry comes close to fulfill the expectations of the "arithmetic site". Fi- nally, we explain how the integral BC-system appears naturally also in de Smit and Lenstra construction of the standard model ofp which singles out the subsystem associated to the ˆ-extension of Q.

Abstract: This comprehensive historical account concerns that non-void intersection region between Riemann zeta function and entire function theory, with a view towards possible physical applications.

Abstract: As usual let s = σ+ it. For any fixed value t = t0 with |t0| ≥ 8, and for σ ≤ 0, we show that |ζ(s)| is strictly monotone decreasing in σ, with the same result also holding for the related functions ξ of Riemann and η of Euler. The following inequality relating the monotonicity of all three functions is proved:

Abstract: We investigate the distribution of the Riemann zeta-function on the line $\Re(s)=\sigma$. For $\tfrac 12 < \sigma \le 1$ we obtain an upper bound on the discrepancy between the distribution of $\zeta(s)$ and that of its random model, improving results of Harman and Matsumoto. Additionally, we examine the distribution of the extreme values of $\zeta(s)$ inside of the critical strip, strengthening a previous result of the first author. As an application of these results we obtain the first effective error term for the number of solutions to $\zeta(s) = a$ in a strip $\tfrac12 < \sigma_1 < \sigma_2 < 1$. Previously in the strip $\tfrac 12 < \sigma < 1$ only an asymptotic estimate was available due to a result of Borchsenius and Jessen from 1948 and effective estimates were known only slightly to the left of the half-line, under the Riemann hypothesis (due to Selberg) and to the right of the abscissa of absolute convergence (due to Matsumoto). In general our results are an improvement of the classical Bohr-Jessen framework and are also applicable to counting the zeros of the Epstein zeta-function.

Abstract: We prove that there is a correspondence between Ramanujan-type formulas for -values. Our method also allows us to reduce certain values of the Epstein zeta function to rapidly converging hypergeometric functions. The Epstein zeta functions were previously studied by Glasser and Zucker.