The adjoint map of the Serre derivative and special values of shifted Dirichlet series

TL;DR

This paper computes the adjoint of the Serre derivative using nearly holomorphic modular forms, linking Fourier coefficients to special values of shifted Dirichlet series, and applies this to derive bounds and formulas for specific functions.

Contribution

It introduces a method to compute the adjoint of the Serre derivative and relates Fourier coefficients to special values of shifted Dirichlet series, providing new insights and formulas.

Findings

Derived an asymptotic bound for special values of shifted Dirichlet series.

Expressed Fourier coefficients in terms of these special values.

Provided a formula for the Ramanujan tau function using shifted Dirichlet series.

Abstract

We compute the adjoint of the Serre derivative map with respect to the Petersson scalar product by using existing tools of nearly holomorphic modular forms. The Fourier coefficients of a cusp form of integer weight $k$, constructed using this method, involve special values of certain shifted Dirichlet series associated with a given cusp form $f$ of weight $k+2$. As application, we get an asymptotic bound for the special values of these shifted Dirichlet series and also relate these special values with the Fourier coefficients of $f$. We also give a formula for the Ramanujan tau function in terms of the special values of the shifted Dirichlet series associated to the Ramanujan delta function.