Dirichlet Series Under Standard Convolutions: Variations on Ramanujan's Identity for Odd Zeta Values

TL;DR

This paper generalizes Ramanujan's identity for Dirichlet series, introducing a formula that linearizes convolutions, leading to new identities and unifying previous results involving Bernoulli numbers.

Contribution

It presents a novel general formula for Dirichlet series convolutions, unifying and extending several known identities related to Bernoulli numbers.

Findings

Derived new identities from the general formula

Recovered known identities as special cases

Unified various convolution identities involving Bernoulli numbers

Abstract

Inspired by a famous identity of Ramanujan, we propose a general formula linearizing the convolution of Dirichlet series as the sum of Dirichlet series with modified weights; its specialization produces new identities and recovers several identities derived earlier in the literature, such as the convolution of squares of Bernoulli numbers by A. Dixit and collaborators, or the convolution of Bernoulli numbers by Y. Komori and collaborators.