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On the Number of Divisors of the Quadratic Form m2 + n2

Published online by Cambridge University Press:  20 November 2018

Gang Yu
Gang Yu*
Affiliation:
Department of Mathematics The University of Georgia Athens, Georgia 30602 U.S.A., e-mail: gyu@alpha.math.uga.edu
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Abstract

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For an integer $n$, let $d\left( n \right)$ denote the ordinary divisor function. This paper studies the asymptotic behavior of the sum

$$S\left( x \right)\,:=\sum\limits_{m\le x,n\le x}{d\left( {{m}^{2}}+{{n}^{2}} \right)}$$
.

It is proved in the paper that, as $x\,\to \,\infty $,

$$S(x):={{A}_{1}}{{x}^{2}}\log x+{{A}_{2}}{{x}^{2}}+{{O}_{\in }}({{x}^{\frac{3}{2}+\in }}),$$

where ${{A}_{1}}$ and ${{A}_{2}}$ are certain constants and $\in $ is any fixed positive real number.

The result corrects a false formula given in a paper of Gafurov concerning the same problem, and improves the error $O({{x}^{\frac{5}{3}}}\,{{(\log \,x)}^{9}})$ claimed by Gafurov.

Keywords

11G0514H52Divisorlarge sieveexponential sums
Type
Research Article
Information
Canadian Mathematical Bulletin , Volume 43 , Issue 2 , 01 June 2000 , pp. 239 - 256
Copyright
Copyright © Canadian Mathematical Society 2000

References

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