Representation of Ramanujan's tau function by twisted divisor functions

TL;DR

This paper introduces a new family of identities expressing Ramanujan's tau function through convolution sums of twisted divisor functions, utilizing modular forms of higher levels.

Contribution

It constructs non-vanishing level 1 cusp forms explicitly from higher level modular forms, providing novel identities for the tau function.

Findings

New identities for Ramanujan's tau function in terms of twisted divisor functions

Explicit construction of non-vanishing level 1 cusp forms from higher level modular forms

Enhanced understanding of the relationship between modular forms and tau function

Abstract

We present an infinite family of identities that represent Ramanujan's tau function in terms of convolution sums of twisted divisor functions. Our method involves explicitly constructing non-vanishing level $1$ cusp forms from modular forms of higher levels.