Abstract. Let K be a p−adic field, and ZΦ(s, f), s ∈ C, with Re(s) > 0, the Igusa local zeta function associated to f(x) = (f1(x), .., fl(x)) ∈ [K (x1, .., xn)] , and Φ a Schwartz-Bruhat function. The aim of this paper is to describe explicitly the poles of the meromorphic continuation of ZΦ(s, f). Using resolution of singularities is possible to express ZΦ(s, f) as a finite sum of p−adic monom...

Observing a multiple version of the divisor function we introduce a new zeta function which we call a multiple finite Riemann zeta function. We utilize some q-series identity for proving the zeta function has an Euler product and then, describe the location of zeros. We study further multi-variable and multi-parameter versions of the multiple finite Riemann zeta functions and their infinite cou...

In this note, we apply the group-theoretic method to study Artin’s conjecture, and introduce the notations of nearly nilpotent groups and nearly supersolvable groups to answer of a question of Arthur and Clozel. As an application, we show that Artin’s conjecture is valid for all nearly supersolvable Galois extensions of number fields as well as all solvable Frobenius extensions.

The aim of this paper is to define an n− 1-cocycle σ on GLn(Q) with values in a certain space D of distributions on Af \ {0}. Here Af denotes the ring of finite adèles of Q, and the distributions take values in the Laurent series C((z1, . . . , zn)). This cocycle can be used to evaluate special values of Artin L-functions on number fields at negative integers. The construction generalizes that ...

Abstract. In this paper we shall develop a theory of (extended) double shuffle relations of Euler sums which generalizes that of multiple zeta values (see Ihara, Kaneko and Zagier, Derivation and double shuffle relations for multiple zeta values. Compos. Math. 142 (2)(2006), 307–338). After setting up the general framework we provide some numerical evidence for our two main conjectures. At the ...

To an arbitrary intersection of exceptional varieties of an embedded resolution we associate a nite number of congruences between naturally occurring multiplicities. This theory generalizes previous results concerning just one exceptional variety. Moreover we describe precise equalities which imply the congruences and we give some applications on the poles of Igusa's local zeta function.

Given an abelian Galois extension K/F of number fields, a quaternion algebra A over F that is ramified at all infinite primes, and a character χ of the Galois group of K over F , we consider the twist of the zeta function of A by the character χ. We show that such twisted zeta functions provide a factorization of the zeta function of A(K) = A ⊗F K. Also, the quotient of the zeta function for A(...

We study Ruelle’s type zeta and L-functions for a torsion free abelian group Γ of rank ν ≥ 2 defined via an Euler product. It is shown that the imaginary axis is a natural boundary of this zeta function when ν = 2, 4 and 8, and in particular, such a zeta function has no determinant expression. Thus, conversely, expressions like Euler’s product for the determinant of the Laplacians of the torus ...

Let φ denote Euler’s phi function. For a fixed odd prime q we investigate the first and second order terms of the asymptotic series expansion for the number of n 6 x such that q ∤ φ(n). Part of the analysis involves a careful study of the Euler-Kronecker constants for cyclotomic fields. In particular, we show that the Hardy-Littlewood conjecture about counts of prime k-tuples and a conjecture o...

We nd a condition for weights on the edges of a graph which insures that the Ihara zeta function has a 3-term determinant formula. Then we investigate the locations of poles of abelian graph coverings and compare the results with random covers. We discover that the zeta function of the random cover satis es an approximate Riemann hypothesis while that of the abelian cover does not.