Abstract: The main objective of this paper is to prove the monodromy conjecture for the local Igusa zeta function of a quasi-ordinary polynomial of arbitrary dimension defined over a number field. In order to do it, we compute the local Denef-Loeser motivic zeta function Z(DL)(h,T) of a quasi-ordinary power series h of arbitrary dimension over an algebraically closed field of characteristic zero from its characteristic exponents without using embedded resolution of singularities. This allows us to effectively represent Z(DL)(h, T) = P(T)/Q(T) such that almost all the candidate poles given by Q(T) are poles. Anyway, these candidate poles give eigenvalues of the monodromy action on the complex R psi(h) of nearby cycles on h(-1)(0). In particular we prove in this case the monodromy conjecture made by Denef-Loeser for the local motivic zeta function and the local topological zeta function. As a consequence, if h is a quasi-ordinary polynomial defined over a number field we prove the Igusa monodromy conjecture for its local Igusa zeta function.

Abstract: This paper investigates the spectral zeta function of the non-commutative harmonic oscillator studied in [PW1, 2]. It is shown, as one of the basic analytic properties, that the spectral zeta function is extended to a meromorphic function in the whole complex plane with a simple pole at s=1, and further that it has a zero at all non-positive even integers, i.e. at s=0 and at those negative even integers where the Riemann zeta function has the so-called trivial zeros. As a by-product of the study, both the upper and the lower bounds are also given for the first eigenvalue of the non-commutative harmonic oscillator.

Abstract: A Brief History of Prime.- Diophantine Equations.- Quadratic Diophantine Equations.- Recovering the Fundamental Theorem of Arithmetic.- Elliptic Curves.- Elliptic Functions.- Heights.- The Riemann Zeta Function.- The Functional Equation of the Riemann Zeta Function.- Primes in an Arithmetic Progression.- Converging Streams.- Computational Number Theory.

Abstract: We show basic properties of zeta functions over the one element field starting from an algebraic set over the integer ring. We calculate several examples and we investigate special values via the associated K-group identified as the stable homotopy group of spheres.

Abstract: The aim of the present work is to exhibit a new proof of the explicit spectral expansion for the fourth moment of the Riemann zeta-function that was established by the second named author a decade ago. Our proof is new, particularly in the sense that it dispenses completely with the Kloostermania, the spectral theory of sums of Kloosterman sums that was used in the former proof. The argument is now constructed precisely upon the spectral structure of the Lie group PSL(2,R). Main ingredients in our argument are the theory of automorphic representations as well as the harmonic analysis on the big Bruhat cell. In essence, this work of ours indicates a new way to view the Riemann zeta-function.

Abstract: We study the special values at s = 2 and 3 of the spectral zeta function ζQ(s) of the non-commutative harmonic oscillator Q(x, Dx) introduced in A. Parmeggiani and M. Wakayama (Proc. Natl Acad. Sci. USA 98 (2001), 26-31; Forum Math. 14 (2002), 539-604). It is shown that the series defining ζQ(s) converges absolutely for Re s > 1 and further the respective values ζQ(2) and ζQ(3) are represented essentially by contour integrals of the solutions, respectively, of a singly confluent Heun ordinary differential equation and of exactly the same but an inhomogeneous equation. As a by-product of these results, we obtain integral representations of the solutions of these equations by rational functions.

Abstract: We give explicit formulae for the local normal zeta functions of torsion-free, class-2-nilpotent groups, subject to conditions on the associated Pfaffian hypersurface which are generically satisfied by groups with small centre and sufficiently large abelianization. We show how the functional equations of two types of zeta functions – the Weil zeta function associated to an algebraic variety and zeta functions of algebraic groups introduced by Igusa – match up to give a functional equation for local normal zeta functions of groups. We also give explicit formulae and derive functional equations for an infinite family of class-2-nilpotent groups known as Grenham groups, confirming conjectures of du Sautoy.

Abstract: One purpose of this paper is to define the twisted q-Bernoulli numbers by using p-adic invariant integrals on Zp. Finally, we construct the twisted q-zeta function and q-L-series which interpolate the twisted q-Bernoulli numbers.

Abstract: We use a smoothed version of the explicit formula to find an approximation to the Riemann zeta function as a product over its nontrivial zeros multiplied by a product over the primes. We model the first product by characteristic polynomials of random matrices. This provides a statistical model of the zeta function that involves the primes in a natural way. We then employ the model in a heuristic calculation of the moments of the modulus of the zeta function on the critical line. This calculation illuminates recent conjectures for these moments based on connections with random matrix theory.

Abstract: Mathematics Subject Classification (2000): Primary 11M26; Secondary 11K38 We continue our investigation of the distribution of the fractional parts of �, whereis a fixed non-zero real number and runs over the imaginary parts of the non-trivial zeros of the Riemann zeta function. We establish some connections to Mont- gomery's pair correlation function and the distribution of primes in short intervals. We also discuss analogous results for a more general L-function.

Abstract: We give an explicit representation for the sums of multiple zeta-star values of fixed weight and height in terms of Riemann zeta values.

Abstract: Introduction A class of real prehomogeneous spaces The orbits of $G$ in $V^+$ The symmetric spaces $G\slash H$ Integral formulas Functional equation of the zeta function for Type I and II Functional equation of the zeta function for Type III Zeta function attached to a representation in the minimal spherical principal series Appendix: The example of symmetric matrices Tables of simple regular graded Lie algebras References Index.

Abstract: The double zeta function was first studied by Euler in response to a letter from Goldbach in 1742. One of Euler's results for this function is a decomposition formula, which expresses the product of two values of the Riemann zeta function as a finite sum of double zeta values involving binomial coefficients. Here, we establish a q-analog of Euler's decomposition formula. More specifically, we show that Euler's decomposition formula can be extended to what might be referred to as a “double q-zeta function” in such a way that Euler's formula is recovered in the limit as q tends to 1.

Abstract: A profinite group G is positively finitely generated (PFG) if for some k, the probability P(G,k) that k random elements generate G is positive. It was conjectured that if G is PFG, then the function P(G,k) can be interpolated to an analytic function defined in some right half-plane. Here that conjecture is formulated more precisely, and verified for (the profinite completion of) arithmetic groups satisfying the congruence subgroup property.

Abstract: I wrote this paper in 1979, as an attempt to extend the results of Borel [2] on zeta functions at negative integers to Artin L-functions. The conceptual framework was provided by Tate’s formulation [10] of Stark’s conjectures. What I needed was a workable definition of the regulator homomorphism in complex K-theory. I discussed this with Borel at the Institute, first over lunch and then in his office. It is an honor to dedicate this paper to his memory. Using results of Bloch and Thurston, I was able to treat the special case of Dirichlet L-series at s = −1. I had the hope of treating Dirichlet L-series at all negative integers, where the order of vanishing is either zero or one, but was unable to construct the required “cyclotomic classes” in K-theory. This was done by Beilinson [11], who also found the generalization of my conjecture, and the conjectures of Deligne [6] on special values, to all motivic L-functionss. I didn’t publish this paper, but it has circulated as a preprint for 25 years. For reasons of historical interest, I decided to publish it in its original form here. I have updated the references, and added some comments on the recent literature at the end of the paper.

Abstract: We introduce a generalized notion of the zeta regularization. As applications we show the quantum analogue of the Lerch formula and of the Dirichlet class number formulas. We also give some further examples related to Appell's O-functions.

Abstract: In this paper we initiate a geometrically oriented construction of non-abelian zeta functions for curves defined over finite fields. More precisely, we first introduce new yet genuine non-abelian zeta functions for curves defined over finite fields, by a "weighted count" on rational points over the corresponding moduli spaces of semi-stable vector bundles using moduli interpretation of these points. Then we define non-abelian L-functions for curves over finite fields using integrations of Eisenstein series associated to L2-automorphic forms over certain generalized moduli spaces. Introduction. In this paper we initiate a geometrically oriented construction of non-abelian zeta functions for curves defined over finite fields. It consists of two chapters. More precisely, in Chapter I, we first introduce new yet genuine non-abelian zeta functions for curves defined over finite fields. This is achieved by a "weighted count" on rational points over the corresponding moduli spaces of semi-stable vector bundles using moduli interpretation of these points. We justify our con- struction by establishing basic properties for these new zetas such as functional equation and rationality, and show that if only line bundles are involved, our newly defined zetas coincide with Artin's Zeta. All this, in particular, the ratio- nality, then leads naturally to our definition of (global) non-abelian zeta functions (for curves defined over number fields), which themselves are justified by a con- vergence result. We end this chapter with a detailed study on rank two non-abelian zeta functions for genus two curves, based on what we call infinitesimal structures of Brill-Noether loci (and Weierstrass points). In Chapter II, we begin with a similar construction for the field of rationals to motivate what follows. In particular, we show that there is an intrinsic relation between our non-abelian zeta functions and Eisenstein series. Due to this, instead of introducing general non-abelian L-functions for curves defined over finite fields with more general test functions (as what Tate did in his Thesis for abelian L- functions), we then define non-abelian L-functions for curves over finite fields as integrations of Eisenstein series associated to L2-automorphic forms over certain generalized moduli spaces. Here geometric truncations play a key role. Basic properties for these non-abelian L-functions, such as meromorphic continuation,

Abstract: In 1999, Iwan Duursma defined the zeta function for a linear code as a generating function of its Hamming weight enumerator. It has various properties similar to those of the zeta function of an algebraic curve. This article extends Duursma's theory to the case of formal weight enumerators. It is shown that the zeta function for a formal weight enumerator has a similar structure to that of the weight enumerator of a Type II code. The notion of the extremal formal weight enumerators is introduced and an analogue of the Mallows-Sloane bound is obtained. Moreover the ternary case is considered.

Abstract: We give an algorithm for computing the special values of twisted standard zeta functions of elliptic modular forms by using the pullback formula for the Siegel-Eisenstein series of degree 2.

Abstract: A recent method of Soundararajan enables one to obtain improved Ω-result for finite series of the form ∑nf(n) cos (2πλnx+β) where 0≤λ1≤λ2≤. . . and β are real numbers and the coefficients f(n) are all non-negative. In this paper, Soundararajan’s method is adapted to obtain improved Ω-result for E(t), the remainder term in the mean-square formula for the Riemann zeta-function on the critical line. The Atkinson series for E(t) is of the above type, but with an oscillating factor (−1)n attached to each of its terms.

Abstract: Several results are obtained concerning multiplicities of zeros of the Riemann zeta-function $\zeta(s)$ They include upper bounds for multiplicities, showing that zeros with large multiplicities have to lie to the left of the line $\sigma = 1$ A zero-density counting function involving multiplicities is also discussed

Abstract: It is believed that the Lindelof hypothesis is also true for the Lerch zeta-function. Here we present results supporting this conjecture. We first consider the growth of the Lerch zeta-function assuming the generalized Lindelof hypothesis for Dirichlet L-functions. We next prove that Huxley’s exponent 32/205 in the Lindelof hypothesis for the Riemann zeta-function holds also for the Lerch zeta-function.

Abstract: We define a generalized Euler gamma function Λβ(z), where the product is taken over powers of integers rather than integers themselves. Studying the associated spectral functions and in particular the zeta function, we obtain the main properties of Λβ(z) and its asymptotic expansion for large values of the argument.

Abstract: Let F2, d denote the free class-2-nilpotent group on d generators. We compute the normal zeta functions prove that they satisfy local functional equations and determine their abscissae of convergence and pole orders.

Abstract: On the space of real rectangular n × m matrices, we introduce a composite power function and study the zeta integral associated with it. We describe the properties of the Igusa zeta function on the basis of the properties of a generalized composite power function and establish a functional relation for the zeta integral. As a result, the Fourier transform of a generalized composite power function is found in explicit form.

Abstract: We give an explicit description of the poles of the Igusa local zeta function associated to a polynomial mapping g, in the case in which it is a nondegenerate homogeneous mapping of degree d. The proof uses a generalization of the p-adic stationary phase formula and Neron p-desingularization.