Journal of the Ramanujan Mathematical Society

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On a Conjecture of Bateman About r5(n)


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1 Department of Mathematics and Statistics, Queen’s University, Kingston, Canada
     

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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|j|≤√n σ(n−j2) 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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  • On a Conjecture of Bateman About r5(n)

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Authors

Arpita Kar
Department of Mathematics and Statistics, Queen’s University, Kingston, Canada
M. Ram Murty
Department of Mathematics and Statistics, Queen’s University, Kingston, Canada

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|j|≤√n σ(n−j2) 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).

References