Special values of shifted convolution Dirichlet series

TL;DR

This paper studies special values of shifted convolution Dirichlet series, showing their generating functions can be expressed as combinations of quasimodular and mock modular forms, advancing understanding of their modular properties.

Contribution

It introduces a framework for expressing generating functions of special values of shifted convolution L-functions as linear combinations of quasimodular and mock modular forms.

Findings

Generating functions are linear combinations of weakly holomorphic quasimodular forms.

They also involve 'mixed mock modular' forms.

Results hold under certain mild conditions.

Abstract

In a recent important paper, Hoffstein and Hulse generalized the notion of Rankin-Selberg convolution $L$-functions by defining shifted convolution $L$-functions. We investigate symmetrized versions of their functions. Under certain mild conditions, we prove that the generating functions of certain special values are linear combinations of weakly holomorphic quasimodular forms and "mixed mock modular" forms.