Abstract: We present a deterministic polynomial time algorithm for computing the zeta function of an arbitrary variety of fixed dimension over a finite field of small characteristic. One consequence of this result is an efficient method for computing the order of the group of rational points on the Jacobian of a smooth geometrically connected projective curve over a finite field of small characteristic. CONTENTS

Abstract: In this series of seven papers, predominantly by means of elementary analysis, we establish a number of identities related to the Riemann zeta function. Whilst this paper is mainly expository, some of the formulae reported in it are believed to be new, and the paper may also be of interest specifically due to the fact that most of the various identities have been derived by elementary methods.

Abstract: In this paper, we consider the relationship between the congruence of cuspidal Hecke eigenforms with respect to Spn(Z) and the special values of their standard zeta functions. In particular, we propose a conjecture concerning the congruence between Saito-Kurokawa lifts and non-Saito-Kurokawa lifts, and prove it under certain condition.

Abstract: Starting with Ihara's work in 1968, there has been a growing interest in the study of zeta functions of finite graphs, by Sunada, Hashimoto, Bass, Stark and Terras, Mizuno and Sato, to name just a few authors. Then, Clair and Mokhtari-Sharghi studied zeta functions for infinite graphs acted upon by a discrete group of automorphisms. The main formula in all these treatments establishes a connection between the zeta function, originally defined as an infinite product, and the Laplacian of the graph. In this article, we consider a different class of infinite graphs. They are fractal graphs, i.e. they enjoy a self-similarity property. We de. ne a zeta function for these graphs and, using the machinery of operator algebras, we prove a determinant formula, which relates the zeta function with the Laplacian of the graph. We also prove functional equations, and a formula which allows approximation of the zeta function by the zeta functions of finite subgraphs.

Abstract: In her Ph.D. Thesis, Czarneski began a preliminary study of the coefficients of the reciprocal of the Ihara zeta function of a finite graph. We give a survey of the results in this area and then give a complete characterization of the coefficients. As an application, we give a (very poor) bound on the number of Eulerian circuits in a graph. We also use these ideas to compute the zeta function of graphs which are cycles with a single chord. We conclude by posing several questions for future work.

Abstract: Recently, Choi et al. (2008) have studied the -extensions of the Apostol-Bernoulli and the Apostol-Euler polynomials of order and multiple Hurwitz zeta function. In this paper, we define Apostol's type -Euler numbers and -Euler polynomials . We obtain the generating functions of and , respectively. We also have the distribution relation for Apostol's type -Euler polynomials. Finally, we obtain -zeta function associated with Apostol's type -Euler numbers and Hurwitz_s type -zeta function associated with Apostol's type -Euler polynomials for negative integers.

Abstract: In this paper, we obtain functional relations for Witten zeta functions by using relations of double Lerch values. By these functional relations, we obtain new proofs of known results on the Tornheim double zeta values, the Euler-Zagier double zeta values, and their alternating and character analogues.

Abstract: Abstract A q-analogue ζ q (s) of the Riemann zeta function ζ(s) was studied in [Kaneko M., Kurokawa N. and Wakayama M.: A variation of Euler's approach to values of the Riemann zeta function. Kyushu J. Math. 57 (2003), 175–192] via a certain q-series of two variables. We introduce in a similar way a q-analogue of the Dirichlet L-functions and make a detailed study of them, including some issues concerning the classical limit of ζ q (s) left open in [Kaneko M., Kurokawa N. and Wakayama M.: A variation of Euler's approach to values of the Riemann zeta function. Kyushu J. Math. 57 (2003), 175–192]. We also examine a “crystal” limit (i.e. q ↓ 0) behavior of ζ q (s). The q-trajectories of the trivial and essential zeros of ζ(s) are investigated numerically when q moves in (0, 1]. Moreover, conjectures for the crystal limit behavior of zeros of ζ q (s), which predict an interesting distribution of “trivial zeros” and an analogue of the Riemann hypothesis for a crystal zeta function, are given. 2000 Mathematics Subject Classification: 11M06.

Abstract: Some nonlocal and nonpolynomial scalar field models originated from p-adic string theory are considered. Infinite number of spacetime derivatives is governed by the Riemann zeta function through d'Alembertian $\Box$ in its argument. Construction of the corresponding Lagrangians begins with the exact Lagrangian for effective field of p-adic tachyon string, which is generalized replacing p by arbitrary natural number n and then taken a sum of over all n. Some basic classical field properties of these scalar fields are obtained. In particular, some trivial solutions of the equations of motion and their tachyon spectra are presented. Field theory with Riemann zeta function nonlocality is also interesting in its own right.

Abstract: The non-commutative harmonic oscillator is a 2×2-system of harmonic oscillators with a non-trivial correlation. We write down explicitly the special value at s=2 of the spectral zeta function of the non-commutative harmonic oscillator in terms of the complete elliptic integral of the first kind, which is a special case of a hypergeometric function.

Abstract: Two new concepts of zeta functions for schemes over the field of one element are proposed. A localization formula and an explicit formula in the affine case are given. This allows for a computation for every scheme.

Abstract: The definition and main properties of the Ihara zeta function for graphs are reviewed, focusing mainly on the case of periodic simple graphs. Moreover, we give a new proof of the associated determinant formula, based on the treatment developed by Stark and Terras for finite graphs.

Abstract: For a cubic extension K3/ℚ, which is not normal, new results on the behavior of mean values of the Dedekind zeta function of the field K3 in the critical strip are obtained.

Abstract: By application of the Markov-WZ method, we prove a more general form of a bivariate generating function identity containing, as particular cases, Koecher's and Almkvist-Granville's Apery-like formulae for odd zeta values. As a consequence, we get a new identity producing Apery-like series for all ζ(2n+4m+3),n,m ≥ 0, convergent at the geometric rate with ratio 2−10.

Abstract: We give an introductory account of the general algorithmic theory of the zeta function of an algebraic set defined over a finite field. CONTENTS

Abstract: We prove a joint universality theorem in the Voronin sense for the periodic Hurwitz zeta-functions.

Abstract: We use a variant of a method of Goncharov, Kontsevich and Zhao [5, 16] to meromorphically continue the multiple Hurwitz zeta function to , to locate the hyperplanes containing its possible poles and to compute the residues at the poles. We explain how to use the residues to locate trivial zeros of .

Abstract: Using WZ-pairs we present simpler proofs of Koecher, Leshchiner and BaileyBorwein-Bradley’s identities for generating functions of the sequences f (2n+2)gn 0 and f (2n + 3)gn 0: By the same method, we give several new representations for these generating functions yielding faster convergent series for values of the Riemann zeta function.

Abstract: In the previous paper [9] the author proved the joint limit theorem for the Riemann zeta function and the Hurwitz zeta function attached with a transcendental real number As a corollary, the author obtained the joint functional independence for these two zeta functions In this paper, we study the joint value distribution for the Riemann zeta function and the Hurwitz zeta function attached with an algebraic irrational number Especially we establish the weak joint functional independence for these two zeta functions

Abstract: This paper is concerned with the arithmetic of the elliptic K3 surface with configuration [1,1,1,12,3*]. We determine the newforms and zeta-functions associated to X and its twists. We verify conjectures of Tate and Shioda for the reductions of X at 2 and 3.

Abstract: The aim of this paper is to show the relation between the zeta function introduced by Stöhr and the Poincaré series of a curve singularity introduced by Campillo, Delgado and Gusein-Zade for the complex case. The interpretation of the Stöhr zeta function in terms of integrals with respect to the (generalized) Euler characteristic over suitable subsets of the ring of functions (following the similar construction made by the previously named authors for subsets of the projectivization of the ring) provides the bridge between both subjects. Let O be a one-dimensional Cohen-Macaulay Noetherian local ring containing a finite field Fq, with maximal ideal m. Let K be the total ring of fractions of O and let O be the integral closure of O in K. Assume the degree ρ := [O/m : Fq] to be finite. Let R be a ring having a large Jacobson radical (i.e., every prime ideal of R containing the Jacobson radical of R is maximal) and which is its own ring of fractions. A subring V = R of R having R as a ring of fractions is called a Manis valuation ring of R if, for every regular element x ∈ R, we have either x ∈ V or x−1 ∈ V . A discrete Manis valuation of R is a surjective map v : R → Z ∪ {∞} such that v(1) = 0, v(0) = ∞ and, for all a, b ∈ R, v(ab) = v(a) + v(b) and v(a + b) ≥ min({v(a), v(b)}). Discrete Manis valuations give rise to discrete Manis valuation rings and conversely (see [Ki-Vi], I-(2.2)). The integral closure O decomposes as a finite intersection of discrete Manis valuation rings (see [Ki-Vi] for more details) O = V1 ∩ . . . ∩ Vr, with associated discrete Manis valuations v1, . . . , vr. For every i ∈ {1, . . . , r}, we denote mi := m(Vi) ∩O, where m(Vi) is the maximal ideal of Vi. 1. Stöhr zeta function. (1.1) In [St] the so-called Stöhr zeta function is defined as: ζ(O, s) := ∑

Abstract: Recently, Smilansky expressed the determinant of the bond scattering matrix of a graph by means of the determinant of its Laplacian. We present another proof for this Smilansky's formula by using some weighted zeta function of a graph. Furthermore, we reprove a weighted version of Smilansky's formula by Bass' method used in the determinant expression for the Ihara zeta function of a graph.