Abstract: Let G be a profinite group. We denote by rn(G) the number of isomorphism classes of irreducible n-dimensional complex continuous representations of G (so that the kernel is open in G). Following [20], we call rn(G) the representation growth function of G. If G is a finitely generated profinite group, then rn(G) < ∞ for every n if and only if G has the property FAb (that is, H/[H, H] is finite for every open subgroup H of G) [1, Proposition 2]. In the case when G is a finitely generated pro-p group, the property FAb is equivalent to the condition that all derived subgroups G are open. In this paper we shall investigate the function

Abstract: The two-fold aim of the paper is to unify and generalize on the one hand the double integrals of Beukers for $\zeta(2)$ and $\zeta(3),$ and those of the second author for Euler's constant $\gamma$ and its alternating analog $\ln(4/\pi),$ and on the other hand the infinite products of the first author for $e$, and of the second author for $\pi$ and $e^\gamma.$ We obtain new double integral and infinite product representations of many classical constants, as well as a generalization to Lerch's transcendent of Hadjicostas's double integral formula for the Riemann zeta function, and logarithmic series for the digamma and Euler beta functions. The main tools are analytic continuations of Lerch's function, including Hasse's series. We also use Ramanujan's polylogarithm formula for the sum of a particular series involving harmonic numbers, and his relations between certain dilogarithm values.

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: of functions F (s) of degree 1 in the extended Selberg class S ]. Precisely, denoting by qF and θF respectively the conductor and the shift of F (s) (see below for definitions) and writing nα = qFα, we proved that for α > 0 the twist F (s, α) has meromorphic continuation to σ > 0, and it has a simple pole at s = 1 − iθF if and only if nα ∈ N and a(nα) 6= 0 (see Theorem 7.1 of [7]). In [7] we exploited such analytic properties in order to characterize the functions of degree 1 in S ]. In particular, we proved that the only functions of degree 1 in the Selberg class S are the Riemann zeta function ζ(s) and the shifted Dirichlet L-functions L(s+ iθ, χ), with θ ∈ R and primitive characters χ. It turns out that Theorem 7.1 of [7] is a special case of a general result for functions in S] of any degree d > 0. To see this, for d > 0, α ∈ R and F ∈ S] with degree d we consider the non-linear twist

Abstract: We obtain formulas for the spectral zeta function of the Laplacian on symmetric finitely ramified fractals, such as the Sierpinski gasket, and a fractal Laplacian on the interval. These formulas contain a new type of zeta function associated with a polynomial (rational functions also can appear in this context). It is proved that this zeta function has a meromorphic continuation to a half plain with poles contained in an arithmetic progression. It is shown as an example that the Riemann zeta function is the zeta functions of a quadratic polynomial, which is associated with the Laplacian on an interval. The spectral zeta function of the Sierpinski gasket is a product of the zeta function of a polynomial and a geometric part; the poles of the former are canceled by the zeros of the latter. A similar product structure was discovered by M.L. Lapidus for self-similar fractal strings.

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: We consider zeta functions with values in the Grothendieck ring of Chow motives. Investigating the lambda-structure of this ring, we deduce a functional equation for the zeta function of abelian varieties. Furthermore, we show that the property of having a rational zeta function satisfying a functional equation is preserved under products.

Abstract: Introduction Prerequisites Outline of Chapters 2 - 8 Elementary Methods Introduction Some Lemmas Two Fundamental Identities Euler's Recurrence for Sigma(n) More Identities Sums of Two Squares Sums of Four Squares Still More Identities Sums of Three Squares An Alternate Method Sums of Polygonal Numbers Exercises Bernoulli Numbers Overview Definition of the Bernoulli Numbers The Euler-MacLaurin Sum Formula The Riemann Zeta Function Signs of Bernoulli Numbers Alternate The von Staudt-Clausen Theorem Congruences of Voronoi and Kummer Irregular Primes Fractional Parts of Bernoulli Numbers Exercises Examples of Modular Forms Introduction An Example of Jacobi and Smith An Example of Ramanujan and Mordell An Example of Wilton: t (n) Modulo 23 An Example of Hamburger Exercises Hecke's Theory of Modular Forms Introduction Modular Group ? and its Subgroup ? 0 (N) Fundamental Domains For ? and ? 0 (N) Integral Modular Forms Modular Forms of Type Mk(? 0(N) chi) and Euler-Poincare series Hecke Operators Dirichlet Series and Their Functional Equation The Petersson Inner Product The Method of Poincare Series Fourier Coefficients of Poincare Series A Classical Bound for the Ramanujan t function The Eichler-Selberg Trace Formula l-adic Representations and the Ramanujan Conjecture Exercises Representation of Numbers as Sums of Squares Introduction The Circle Method and Poincare Series Explicit Formulas for the Singular Series The Singular Series Exact Formulas for the Number of Representations Examples: Quadratic Forms and Sums of Squares Liouville's Methods and Elliptic Modular Forms Exercises Arithmetic Progressions Introduction Van der Waerden's Theorem Roth's Theorem t 3 = 0 Szemeredi's Proof of Roth's Theorem Bipartite Graphs Configurations More Definitions The Choice of tm Well-Saturated K-tuples Szemeredi's Theorem Arithmetic Progressions of Squares Exercises Applications Factoring Integers Computing Sums of Two Squares Computing Sums of Three Squares Computing Sums of Four Squares Computing rs(n) Resonant Cavities Diamond Cutting Cryptanalysis of a Signature Scheme Exercises References Index

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: We substantially apply the Li criterion for the Riemann hypothesis to hold. Based upon a series representation for the sequence {λk}, which are certain logarithmic derivatives of the Riemann xi function evaluated at unity, we determine new bounds for relevant Riemann zeta function sums and the sequence itself. We find that the Riemann hypothesis holds if certain conjectured properties of a sequence ηj are valid. The constants ηj enter the Laurent expansion of the logarithmic derivative of the zeta function about s=1 and appear to have remarkable characteristics. On our conjecture, not only does the Riemann hypothesis follow, but an inequality governing the values λn and inequalities for the sums of reciprocal powers of the nontrivial zeros of the zeta function.

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 prove that the zeta-function $\zeta_\Delta$ of the Laplacian $\Delta$ on a self-similar fractals with spectral decimation admits a meromorphic continuation to the whole complex plane. We characterise the poles, compute their residues, and give expressions for some special values of the zeta-function. Furthermore, we discuss the presence of oscillations in the eigenvalue counting 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: This paper investigates a new family of special functions referred to as hypergeometric zeta functions. Derived from the integral representation of the classical Riemann zeta function, hypergeometric zeta functions exhibit many properties analogous to their classical counterpart, including the intimate connection to Bernoulli numbers. These new properties are treated in detail and are used to demonstrate a functional inequality satisfied by second-order hypergeometric zeta functions.

Abstract: The paper studies the local zero spacings of deformations of the Riemann ξ-function under certain averaging and differencing operations. For real h we consider the entire functions Ah(s) := 1 (ξ(s + h) + ξ(s − h)) and Bh(s) = 1 2i (ξ(s + h) − ξ(s − h)) . For |h| ≥ 1 2 the zeros of Ah(s) and Bh(s) all lie on the critical line ℜ(s) = 1 and are simple zeros. The number of zeros of these functions to height T has asymptotically the same density as the Riemann zeta zeros. For fixed |h| ≥ 1 the distribution of normalized zero spacings of these functions up to height T converge as T → ∞ to a limiting distribution, which consists of equal spacings of size 1. That is, these zeros are asymptotically regularly spaced. Assuming the Riemann hypothesis, the same properties hold for all nonzero h. In particular, these averaging and differencing operations destroy the (conjectured) GUE distribution of the zeros of the ξ-function, which should hold at h = 0. Analogous results hold for all completed Dirichlet L-functions ξχ(s) having χ a primitive character.

Abstract: In upcoming papers by Conrey, Farmer and Zirnbauer there appear conjectural formulas for averages, over a family, of ratios of products of shifted L-functions. In this paper we will present various applications of these ratios conjectures to a wide variety of problems that are of interest in number theory, such as lower order terms in the zero statistics of L-functions, mollified moments of L-functions and discrete averages over zeros of the Riemann zeta function. In particular, using the ratios conjectures we easily derive the answers to a number of notoriously difficult computations.

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.