Eisenstein Series and Convolution 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 ${\sum }_{a+pb=n}\sigma (a)\sigma (b)$, ${\sum }_{p_{1}a+p_{2}b=n}\sigma (a)\sigma (b)$ and ${\sum }_{a+p_1{p}_{2}b=n}\sigma (a)\sigma (b)$ where $p,p_1,p_2$ are primes.

14 pages

References in corpus (2)

• Evaluating convolutions of divisor sums with quasimodular forms
• A family of Eta Quotients and an Extension of the Ramanujan-Mordell Theorem