On an Arithmetical Function

Abstract

In this paper we introduce an arithmetical function δ(n), the difference between the number of divisors of n congruent to 1 mod 3 and those congruent to −1 mod 3. This function then is related to the classical function σ(n) which is the sum of the divisors of n. In particular we prove the identity

$$3\bigg(\sum_{n=0}^{\infty}\delta(3n+1)x^n\bigg)^2=\sum_{n=0}^{\infty}\sigma(3n+2)x^n.$$