On a certain identity for the cotangent zeta finite Dirichlet series and its application to the Berndt--Arakawa formula

TL;DR

This paper presents a new, simple proof of the Berndt--Arakawa formula for special values of the cotangent zeta function, using an asymptotic identity for finite cotangent Dirichlet series.

Contribution

It introduces a novel method to evaluate special values of infinite Dirichlet series via asymptotic identities of their finite partial sums.

Findings

Proves a key asymptotic identity for finite cotangent Dirichlet series.

Derives the Berndt--Arakawa formula directly from the asymptotic identity.

Suggests a new approach for evaluating special values of Dirichlet series.

Abstract

The cotangent zeta function is a very interesting object, which is related to partial zeta functions and Hecke $L$-functions of real quadratic fields. Its special values at odd integers greater than 1 are explicitly evaluated by Berndt in the real quadratic unit case. Later Arakawa generalized the formula to the arbitrary real quadratic number case. The purpose of this article is to give a novel and surprisingly simple proof of the Berndt--Arakawa formula. Namely we prove a certain asymptotic identity for two finite cotangent Dirichlet series, from which we deduce the Berndt--Arakawa formula immediately. The authors believe that the method employed here of evaluating special values of infinite Dirichlet series from an asymptotic identity for its finite partial sum is of interest in its own right.