Abstract: We provide a lower bound for the dimension of the vector space spanned by 1 and by the values of the Riemann Zeta function at the first odd integers. As a consequence, the Zeta function takes infinitely many irrational values at odd integers.

Abstract: We give a simple combinatorial proof of a formula that extends a result by Grigorchuk (rediscovered by Cohen) relating cogrowth and spectral radius of random walks. Our main result is an explicit equation determining the number of `bumps' on paths in a graph: in a $d$-regular (not necessarily transitive) non-oriented graph let the series $G(t)$ count all paths between two fixed points weighted by their length $t^{length}$, and $F(u,t)$ count the same paths, weighted as $u^{number of bumps}t^{length}$. Then one has $$F(1-u,t)/(1-u^2t^2) = G(t/(1+u(d-u)t^2))/(1+u(d-u)t^2).$$ We then derive the circuit series of `free products' and `direct products' of graphs. We also obtain a generalized form of the Ihara-Selberg zeta function.

Abstract: The Dirichlet class number formula expresses the residue at s = 1 of the Dedekind zeta function ζ F(s) of an arbitrary algebraic number field F as the product of a simple factor (involving the class number of the field) with the determinant of a matrix whose entries are logarithms of units in the field. On the other hand, if F is a totally real number field of degree n, then a famous theorem by Klingen and Siegel says that the value ζ F (m) for every positive even integer in is a rational multiple of π mn In [52] and [53], a conjectural generalization of these two results was formulated according to which the special value ζ F (m) for arbitrary number fields F and positive integers m can be expressed in terms of special values of a transcendental function depending only on m, namely the m th classical polylogarithm function. These instances are expected to form part of a much more general picture in which a special value of an L-series of “motivic origin” is expressed in terms of some transcendental function. In this survey we collect some pieces fitting into and illustrating this picture.

Abstract: In this paper we introduce some new methods to understand the analytic behaviour of the zeta function of a group. We can then combine this knowledge with suitable Tauberian theorems to deduce results about the growth of subgroups in a nilpotent group. In order to state our results we introduce the following notation. For \alpha a real number and N a nonnegative integer, define s_N^\alpha(G) = sum_{n=1}^N a_n(G)/n^\alpha. Main Theorem: Let G be a finitely generated nilpotent infinite group. (1) The abscissa of convergence \alpha(G) of \zeta_G(s) is a rational number and \zeta_G(s) can be meromorphically continued to Re(s)>\alpha(G)-\delta for some \delta >0. The continued function is holomorphic on the line \Re(s) = (\alpha)G except for a pole at s=\alpha(G). (2) There exist a nonnegative integer b(G) and some real numbers c,c' such that s_{N}(G) ~ c N^{\alpha(G)}(\log N)^{b(G)} s_{N}^{\alpha(G)}(G) ~ c' (\log N)^{b(G)+1} for N\rightarrow \infty .

Abstract: To study the distribution of pairs of zeros of the Riemann zeta-function, Montgomery introduced the function F(\alpha) = F_T(\alpha) = \left({T\over 2\pi}\log T\right)^{-1} \sum_{0<\gamma,\gamma ' \le T} T^{i\alpha(\gamma -\gamma ')}w(\gamma-\gamma '), where and denote the imaginary parts of zeros of the Riemann zeta-function, and . Assuming the Riemann Hypothesis, Montgomery proved an asymptotic formula for when , and made the conjecture that for any bounded . In this paper we use an approximation for the prime indicator function together with a new mean value theorem for long Dirichlet polynomials and tails of Dirichlet series to prove that, assuming the Generalized Riemann Hypothesis for all Dirichlet -functions, then for any we have uniformly for and all .1991 Mathematics Subject Classification: primary 11M26; secondary 11P32.

Abstract: 0 f(x)x s 1 dx with s = + it denote the Mellin transform of f(x). Mellin transforms play a fundamental role in Analytic Number Theory. They can be viewed, by a change of variable, as special cases of Fourier transforms, and their properties can be deduced from the general theory of Fourier transforms. For an extensive account, we refer the reader to E. C. Titchmarsh (25). One of the basic properties of Mellin transforms is the inversion formula 1 2 {f(x + 0) +f(x 0)} = 1 2i ( ) F (s)x s ds = 1 2i lim T!1 +iT

Abstract: The purpose of the present article is to survey some mean value results obtained recently in zeta-function theory. We do not mention other important aspects of the theory of zeta-functions, such as the distribution of zeros, value-distribution, and applications to number theory. Some of them are probably treated in the articles of Professor Apostol and Professor Ramachandra in the present volume.

Abstract: Explicit formulas for the zeta functions $\zeta_\alpha (s)$ corresponding to bosonic ($\alpha =2$) and to fermionic ($\alpha =3$) quantum fields living on a noncommutative, partially toroidal spacetime are derived. Formulas for the most general case of the zeta function associated to a quadratic+linear+constant form (in {\bf Z}) are obtained. They provide the analytical continuation of the zeta functions in question to the whole complex $s-$plane, in terms of series of Bessel functions (of fast, exponential convergence), thus being extended Chowla-Selberg formulas. As well known, this is the most convenient expression that can be found for the analytical continuation of a zeta function, in particular, the residua of the poles and their finite parts are explicitly given there. An important novelty is the fact that simple poles show up at $s=0$, as well as in other places (simple or double, depending on the number of compactified, noncompactified, and noncommutative dimensions of the spacetime), where they had never appeared before. This poses a challenge to the zeta-function regularization procedure.

Abstract: This survey article, intended to be accessible to stu- dents, discusses results about the presence or absence of zeroes of the Bergman kernel function of a bounded domain in C n . Six open problems are stated. For example, the series P 1=0 z n =n! is zero-free, but how can one tell this without a priori knowledge that the series represents the exponential function? Changing the initial term of this series produces a new series that does have zeroes, since the range of the exponential function is all non-zero complex numbers. The problem of determining when a series has zeroes is essentially equivalent to the hard problem of determining the range of a holomorphic function that is presented as a series. A famous instance of the problem of locating zeroes of innite series is the Riemann hypothesis about the zeta function: namely, the conjecture that when 0 < Re z< 1 =2, the convergent series P 1=1 ( 1) n =n z has no zeroes. This formulation of the Riemann hypothesis is equivalent to the usual statement that the zeroes of (z) in the critical strip where

Abstract: The theory of geometric zeta functions for locally symmetric spaces is generalized to the case of higher rank spaces. We show that the zeta functions can be continued to meromorphic functions on the plane, describe the divisor in terms of tangential cohomology and in terms of group cohomology which generalizes a conjecture of Patterson. We also extend the range of zeta functions in considering higher dimensional flats.

Abstract: Fejer monotonicity and weak convergence of an accelerated method of projections by H. H. Bauschke, F. Deutsch, H. Hundal, and S.-H. Park General "squeeze theorems" in nonsmooth analysis by J. Benoist and J.-B. Hiriart-Urruty Ramanujan's short unpublished manuscript on integrals and series related to Euler's constant by B. C. Berndt and D. C. Bowman An efficient algorithm for the Riemann zeta function by P. Borwein On periods of elements from real quadratic number fields by E. B. Burger and A. J. van der Poorten Vector-valued perturbed minimization principles by R. Deville and C. Finet Continued fractions, comparison algorithms, and fine structure constants by P. Flajolet and B. Vallee A class of dynamical systems with a convex Lyapunov function by M. Gabour, S. Reich, and A. J. Zaslavski On a convergence of lower semicontinuous functions linked with the graph convergence of their subdifferentials by M. Geoffroy and M. Lassonde Rotundity related to Lipschitz separation by J. R. Giles and J. D. Vanderwerff Codirectional compactness, metric regularity and subdifferential calculus by A. Ioffe Complete dual characterizations of optimality for convex semidefinite programming by V. Jeyakumar and M. J. Nealon A second-welfare theorem in nonconvex economies by A. Jofre Auxiliary problem and the approximation of variational inequalities with non-symmetric multi-valued operators by A. Kaplan and R. Tichatschke Dynamic emission tomography-Regularization and inversion by J. Maeght, D. Noll, and S. Boyd Semi-spaces and bases on convexity systems in complete lattices by J. E. Martinez-Legaz and I. Singer On variational characterizations of Asplund spaces by B. S. Mordukhovich and B. Wang Transposition of relations by J.-P. Penot On a classical existence theorem for nonlinear elliptic equations by B. Ricceri Limiting convex subdifferential calculus with applications to integration and maximal monotonicity of subdifferential by L. Thibault.

Abstract: Spectral zeta functions ζ(s) for the massless scalar fields obeying the Dirichlet and Neumann boundary conditions on a surface of an infinite cylinder are constructed. These functions are defined explicitly in a finite domain of the complex plane s containing the closed interval of real axis −1⩽Re s⩽0. Proceeding from this the spectral zeta functions for the boundary conditions given on a circle (boundary value problem on a plane) are obtained without any additional calculations. The Casimir energy for the relevant field configurations is deduced.

Abstract: A class of dynamical systems associated to rings of S-integers in rational function fields is described. General results about these systems give a rather complete description of the well-known dynamics in one-dimensional additive cellular automata with prime alphabet, including simple formulae for the topological entropy and the number of periodic configurations. For these systems the periodic points are uniformly distributed along some subsequence with respect to the maximal measure, and in particular are dense. Periodic points may be constructed arbitrarily close to a given configuration, and rationality of the dynamical zeta function is characterized. Throughout the emphasis is to place this particular family of cellular automata into the wider context of S-integer dynamical systems, and to show how the arithmetic of rational function fields determines their behaviour. Using a covering space the dynamics of additive cellular automata are related to a form of hyperbolicity in completions of rational function fields. This expresses the topological entropy of the automata directly in terms of volume growth in the covering space.

Abstract: Euler discovered a recursion formula for the Riemann zeta function evaluated at the even integers. He also evaluated special Dirichlet series whose coefficients are the partial sums of the harmonic series. This paper introduces a new method for deducing Euler's formulas as well as a host of new relations, not only for the zeta function but for several allied functions.

Abstract: Let X be a nonsingular algebraic variety in characteristic zero. To an effective divisor on X Kontsevich has associated a certain 'motivic integral', living in a completion of the Grothendieck ring of algebraic varieties. He used this invariant to show that birational Calabi-Yau varieties have the same Hodge numbers. Then Denef and Loeser introduced the motivic (Igusa) zeta function, associated to a regular function on X, which specializes to both the classical p-adic Igusa zeta function and the topological zeta function, and also to Kontsevich's invariant. This paper treats a generalization to singular varieties. Batyrev already considered such a 'Kontsevich invariant' for log terminal varieties (on the level of Hodge polynomials instead of in the Grothendieck ring), and previously we introduced a motivic zeta function on normal surface germs. Here on any Q-Gorenstein variety X we associate a motivic zeta function and a 'Kontsevich invariant' to effective Q-Cartier divisors on X whose support contains the singular locus of X.

Abstract: This paper gives an explicit zero-free region for the Riemann zeta-function derived from the VinogradovKorobov method. We prove that the Riemann zeta-function does not vanish in the region σ ≥ 1 − .00105 log−2/3 |t| (log log |t|)−1/3 and |t| ≥ 3.

Abstract: Many interesting families of rapidly convergent series representations for the Riemann Zeta function $\zeta (2n+1)$ $(n\in {\Bbb N})$ were considered recently by various authors In this survey-cum-expository paper, the author presents a systematic (and historical) investigation of these series representations Relevant connections of the results presented here with several other known series representations for $\zeta (2n+1)$ $(n\in {\Bbb N})$ are also pointed out In one of many computationally useful special cases presented here, it is observed that $\zeta (3)$ can be represented by means of a series which converges much faster than that in Euler's celebrated formula as well as the series used recently by Ap\'{e}ry in his proof of the irrationality of $\zeta (3)$ Symbolic and numerical computations using {\em Mathematica} (Version 40) for Linux show, among other things, that only 50 terms of this series are capable of producing an accuracy of seven decimal places

Abstract: The Laplace transform of $|\zeta(1/2+it)|$ is investigated, for which a precise expression is obtained, valid in a certain region in the complex plane. The method of proof is based on complex integration and spectral theory of the non-Euclidean Laplacian.

Abstract: In this article we describe some ways to significantly improve the Markov-Gauss-Camp-Meidell inequalities and provide specific applications. We also describe how the improved bounds are extendable to the multivariate case. Applications include explicit finite sample construction of confidence intervals for a population mean, upper bounds on a tail probability P(X>k) by using the density at k, approximation of P-values, simple bounds on the Riemann Zeta function, on the series , improvement of Minkowski moment inequalities, and construction of simple bounds on the tail probabilities of asymptotically Poisson random variables. We also describe how a game theoretic argument shows that our improved bounds always approximate tail probabilities to any specified degree of accuracy.

Abstract: Among the thousands of discoveries made by mathematicians over the centuries, some stand out as significant landmarks. One of these is the prime number theorem, which describes the asymptotic distribution of prime numbers. It can be stated in various equivalent forms, two of which are: $$\pi \left( x \right) \sim \frac{x} {{\log x}}as\quad x \to \infty ,$$ (1) and $${p_n} \sim \,n\,\log \,n\,as\,n\, \to \infty.$$ (2)

Abstract: Over four hundred years ago, Sir Walter Raleigh asked his mathematical assistant to find formulas for the number of cannonballs in regularly stacked piles. These investigations aroused the curiosity of the astronomer Johannes Kepler and led to a problem that has gone centuries without a solution: why is the familiar cannonball stack the most efficient arrangement possible? Here we discuss the solution that Hales found in 1998. Almost every part of the 282-page proof relies on long computer verifications. Random matrix theory was developed by physicists to describe the spectra of complex nuclei. In particular, the statistical fluctuations of the eigenvalues (“the energy levels”) follow certain universal laws based on symmetry types. We describe these and then discuss the remarkable appearance of these laws for zeros of the Riemann zeta function (which is the generating function for prime numbers and is the last special function from the last century that is not understood today.) Explaining this phenomenon is a central problem. These topics are distinct, so we present them separately with their own introductory remarks.

Abstract: A zeta function is an analytic function whose analytic properties somehow encapsulate a tremendous amount of arithmetic information. These functions are an uncannily powerful tool in number theory; to mention just some celebrated examples, they are at the heart of the proofs of the Prime Number Theorem, Dirichlet’s theorem on primes in arithmetic progressions, and the main theorems (in their original form) of class field theory, not to mention their more recent incarnation in the Weil conjectures.