Shin-ya Koyama

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)

Multiple sine functions

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

Absolute zeta functions

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