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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