The values of an Euler sum at negative integers and relation to a convolution of Bernoulli numbers

TL;DR

This paper investigates a specific Dirichlet series at negative integers, revealing its connection to convolutions of Bernoulli numbers, thereby extending understanding of special values of Euler sums.

Contribution

It precisely determines the values of a known Dirichlet series at negative integers and links these to convolutions of Bernoulli numbers, providing new insights into their relationships.

Findings

Values of the Dirichlet series at negative integers are explicitly computed.

Established a novel connection between Euler sums and convolutions of Bernoulli numbers.

Extended previous results by Apostol and Matsuoka on special series values.

Abstract

We study a special Dirichlet series studied before by Apostol and Matsuoka and specify its values at negative integers. These values are related to a certain convolution of Bernoulli numbers