The Euler Product expressions of the absolute tensor products of the Dirichlet $L$-functions

TL;DR

This paper derives Euler product formulas for the absolute tensor square of Dirichlet L-functions, generalizing Akatsuka's theorem and confirming Kurokawa's 1992 prediction.

Contribution

It provides a novel Euler product expression for the absolute tensor square of Dirichlet L-functions, extending previous results on the Riemann zeta function.

Findings

Euler product expression for the absolute tensor square of Dirichlet L-functions

Generalization of Akatsuka's theorem

Proof of Kurokawa's 1992 prediction

Abstract

In this paper, we calculate the absolute tensor square of the Dirichlet $L$-functions and show that it is expressed as an Euler product over pairs of primes. The method is to construct an equation to link primes to a series which has the factors of the absolute tensor product of the Dirichlet $ L $-functions. This study is a generalization of Akatsuka's theorem on the Riemann zeta function, and gives a proof of Kurokawa's prediction proposed in 1992.