The space of convolution identities on divisor functions

Abstract

We consider convolutions of divisor functions in arbitrary length with modular congruence restrictions, and introduce the notion of a space of convolution identities over the rational numbers. As a main result, we introduce a conjecture on the connection between the dimension of the space of convolution identities and the number of partitions of a positive integer into exactly three parts, and prove the conjecture for 15 cases. We also prove convolution identities in arbitrary length.