Dirichlet series of Rankin-Cohen Brackets

TL;DR

This paper explores the Dirichlet series associated with Rankin-Cohen brackets of modular forms, expressing them through a correspondence with quasimodular forms and derivatives, revealing new analytical relationships.

Contribution

It introduces a novel method to represent Dirichlet series of derivatives of modular forms using Rankin-Cohen brackets and quasimodular form correspondence.

Findings

Expressed Dirichlet series as linear combinations of Rankin-Cohen brackets

Established a link between quasimodular forms and derivatives of modular forms

Provided new analytical tools for studying modular form products

Abstract

Given modular forms $f$ and $g$ of weights $k$ and $\ell$, respectively, their Rankin-Cohen bracket $[f,g]^{(k, \ell)}_n$ corresponding to a nonnegative integer $n$ is a modular form of weight $k +\ell +2n$, and it is given as a linear combination of the products of the form $f^{(r)} g^{(n-r)}$ for $0 \leq r \leq n$. We use a correspondence between quasimodular forms and sequences of modular forms to express the Dirichlet series of a product of derivatives of modular forms as a linear combination of the Dirichlet series of Rankin-Cohen brackets.