Dualities for absolute zeta functions and multiple gamma functions

TL;DR

This paper explores the normalization of absolute zeta functions using multiple gamma and sine functions, revealing a connection between their functional equations and the properties of non-classical multiple sine functions.

Contribution

It introduces the absolute Hurwitz zeta function and links the theory of multiple gamma and sine functions to the normalization of absolute zeta functions.

Findings

Normalizations via multiple gamma and sine functions are effective in certain cases.

The functional equation of the absolute zeta function relates to the simplicity of non-classical multiple sine functions.

The theory applies to cases involving the Kurokawa tensor product.

Abstract

We study absolute zeta functions from the view point of a canonical normalization. We introduce the absolute Hurwitz zeta function for the normalization. In particular, we show that the theory of multiple gamma and sine functions gives good normalizations in cases related to the Kurokawa tensor product. In these cases, the functional equation of the absolute zeta function turns out to be equivalent to the simplicity of the associated non-classical multiple sine function of negative degree.