Nobushige Kurokawa

Higher Mahler measures and zeta functions

We consider a generalization of the Mahler measure of a multi- variable polynomialP as the integral of log k jPj in the unit torus, as opposed to the classical denition with the integral of log jPj. A zeta Mahler mea- sure, involving the integral ofjPj s , is also considered. Specic examples are

Absolute tensor products

We calculate the absolute tensor product ζ(s,Fp1)⊗···⊗ζ(s,Fpr) by observing a relation between the multiple sine function of Shintanis type and Appells O-function, or equivalently, the multiple elliptic gamma function. This allows us to give an Euler product expression for an absolute tensor product.

Euler Products Beyond the Boundary

We investigate the behavior of the Euler products of the Riemann zeta function and Dirichlet L-functions on the critical line. A refined version of the Riemann hypothesis, which is named “the Deep Riemann Hypothesis”, is examined. We also study various analogs for global function fields. We give an interpretation for the nontrivial zeros from the viewpoint of statistical mechanics.

Multiple zeta functions: the double sine function and the signed double Poisson summation formula

We construct multiple zeta functions as absolute tensor products of usual zeta functions. The Euler product expression is established for the most basic case

Euler's integrals and multiple sine functions

\zeta(s,\mathbf{F}_p)\otimes\zeta(s,\mathbf{F}_q)

Casimir effects on Riemann surfaces

by using the signed double Poisson summation formula and the theory of the double sine function.

On q-analogues of the Euler constant and Lerch's limit formula

We show that Eulers famous integrals whose integrands contain the logarithm of the sine function are expressed via multiple sine functions.

On q-Basic Multiple Gamma Functions

Abstract The Casimir effect whose existence was first predicted by Casimir in 1948 is considered as a manifestation of macroscopic quantum field theory. This force is evaluated theoretically by using the value of the Riemann zeta function at −3. The aim of the present paper is to introduce a similar Casimir energy for a Riemann surface, and to express it by a special value of the Mellin transform of a theta series arising from the heat kernel and also by a weighted integral of the logarithm of the Selberg zeta function.

Analyticity of polylogarithmic Euler products

We introduce and study a q-analogue γ(q) of the Euler constant via a suitably defined q-analogue of the Riemann zeta function. We show, in particular, that the value γ(2) is irrational. We also present a q-analogue of the Hurwitz zeta function and establish an analogue of the limit formula of Lerch in 1894 for the gamma function. This limit formula can be regarded as a natural generalization of the formula of γ(q).

JACKSON'S INTEGRAL OF THE HURWITZ ZETA FUNCTION

We define a q-analogue of the basic multiple gamma function introduced in [17] which differs from the one defined by Barnes [1] via the zeta regularized product. We call it a q-basic multiple gamma function. Using this q-basic multiple Gamma function we introduce a q-analogue of the multiple sine function of order m + 1. We study properties of such functions from the periodicity, differential-difference equations and duplication formulas, etc. points of view.