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

2021/07/14 by Chavan, Parth, Chavan, Sarth, Vignat, Christophe +1
#FOS: Mathematics #Number Theory (math.NT) #Primary: 11Mxx #Secondary: 11B68 and 11F03

paper · doi:10.48550/arxiv.2107.06457

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.

Related

• Recent developments pertaining to Ramanujan's formula for odd zeta values
• Applications of Ramanujan's work on Eisenstein series
• Sums with convolution of Dirichlet characters
• Asymptotic identities for additive convolutions of sums of divisors
• An elementary proof of some Ramanujan-type identities