Eisenstein Series and Convolution Sums

TL;DR

This paper computes Fourier expansions of Eisenstein series at various cusps and uses these to derive formulas for specific convolution sums involving divisor functions and primes.

Contribution

It introduces explicit formulas for convolution sums using Fourier expansions of Eisenstein series at different cusps, connecting modular forms to number theory.

Findings

Formulas for convolution sums involving divisor functions and primes

Explicit Fourier series expansions at various cusps

Connections between Eisenstein series and number theoretic sums

Abstract

We compute Fourier series expansions of weight $2$ and weight $4$ Eisenstein series at various cusps. Then we use results of these computations to give formulas for the convolution sums $ \sum_{a+p b=n}\sigma(a)\sigma(b)$, $ \sum_{p_1a+p_2 b=n}\sigma(a)\sigma(b)$ and $ \sum_{a+p_1 p_2 b=n}\sigma(a)\sigma(b)$ where $p, p_1, p_2$ are primes.