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Journal of the Ramanujan Mathematical Society

Volume 34, Issue 1, March 2019  pp. 133–142.

On a conjecture of Bateman about r 5 (n)

Authors:  Arpita Kar and M. Ram Murty
Author institution:Department of Mathematics and Statistics, Queen's University, Kingston, ON Canada K7L 3N6

Summary:  Let r 5(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 ∑ |j| ≤ √ n σ (n - j 2) 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).

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