Infinite products involving Dirichlet characters and cyclotomic polynomials

TL;DR

This paper evaluates infinite products involving Dirichlet characters and cyclotomic polynomials using gamma functions and sines, leading to explicit formulas and evaluations of multiple L-series.

Contribution

It introduces new methods to express infinite products of Dirichlet characters and cyclotomic polynomials as finite products of gamma functions and sines.

Findings

Explicit formulas for infinite products involving Dirichlet characters.

Finite product representations of cyclotomic polynomial products.

Evaluations of certain multiple L-series.

Abstract

Using some basic properties of the gamma function, we evaluate a simple class of infinite products involving Dirichlet characters as a finite product of gamma functions and, in the case of odd characters, as a finite product of sines. As a consequence we obtain evaluations of certain multiple $L$-series. In the final part of this paper we derive expressions for infinite products of cyclotomic polynomials, again as finite products of gamma or of sine functions.