In this lecture we give an introduction to the Grothendieck ring of algebraic varieties, and discuss Kapranov’s lifting of the Hasse-Weil zeta function to this Grothendieck ring. One interesting feature is that this makes sense over an arbitrary field. We will prove the rationality of Kapranov’s zeta function for curves by a variant of the argument used in Lecture 4 for the Hasse-Weil zeta func...

We compute the equivariant zeta function for bundles over infinite graphs and for infinite covers. In particular, we give a “transfer formula” for the zeta function of infinite graph covers. Also, when the infinite cover is given as a limit of finite covers, we give a formula for the limit of the zeta functions.

The argument of a zeta function is a complex number. We can interpret a complex number as the subgroup of a quaternion. Therefore, we can expand the argument of a zeta function to a quaternion. On the other hand, a complex number is one-dimensional complex general linear group. And the sporadic finite simple groups are high dimensional complex general linear groups. Therefore, we can expand the...

INTRODUCTION 4 Introduction Zeta functions encoding geometric information such as zeta functions of algebraic varieties over finite fields or zeta functions of finite graphs will loosely be called geometric zeta functions in the sequel. Sometimes the geometric situation gives one tools at hand to prove analytical continuation, functional equation and an adapted form of the Riemann hypothesis. T...

We will define the extension of q-Hurwitz zeta function due to Kim and Rim (2000) and study its properties. Finally, we lead to a useful new integral representation for the q-zeta function. 1. Introduction. Let 0 < q < 1 and for any positive integer k, define its q-analogue [k] q = (1 − q k)/(1 − q). Let C be the field of complex numbers. The q-zeta function due

In this brief note we bring out the analogy between the arithmetic of elliptic curves and the Riemann zeta-function. · · · The aim of this note is to point out the possibility of developing the theory of elliptic curves so that the arithmetical and analytic aspects are developed in strict analogy with the classical theory of the Riemann zeta-function. This has been triggered by Lemmermeyer’s pe...

This series is absolutely convergent in the region Rs > 1, and defines a holomorphic function in s there. We call this function ζQ(s) the spectral zeta function for the non-commutative harmonic oscillator Q, which is introduced and studied by Ichinose and Wakayama [1]. The zeta function ζQ(s) is analytically continued to the whole complex plane as a single-valued meromorphic function which is h...

The L-zeta function of an infinite graph Y (defined previously in a ball around zero) has an analytic extension. For a tower of finite graphs covered by Y , the normalized zeta functions of the finite graphs converge to the L-zeta function of Y . Introduction Associated to any finite graph X there is a zeta function Z(X,u), u ∈ C. It is defined as an infinite product but shown (in various diffe...

In this article, we introduce a systematic new method to investigate the conjectural p-adic meromorphic continuation of Professor Bernard Dwork’s unit root zeta function attached to an ordinary family of algebraic varieties defined over a finite field of characteristic p. After his pioneer p-adic investigation of the Weil conjectures on the zeta function of an algebraic variety over a finite fi...