On a Conjecture of Bateman About r5 (n)

Abstract

Let r5(n) be the number of ways of writing n as a sum of five integer squares. In his study of this function, Bateman was led to formulate a conjecture regarding the sum Σ σ(n − j^2) | j|≤√n where σ (n) is the sum of positive divisors of n. We give a proof of Bateman’s conjecture in the case n is square-free and congruent to 1 (mod 4).