On the Number of Divisors of the Quadratic Form $m^2+n^2$

Canadian mathematical bulletin

关键词: divisor;    large sieve;    exponential sums;   

DOI  :  10.4153/CMB-2000-032-3

学科分类：数学（综合）

来源: University of Toronto Press * Journals Division

【 摘 要 】

For an integer $n$, let $d(n)$ denote the ordinary divisor function.This paper studies the asymptotic behavior of the sum$$S(x) := sum_{mleq x, nleq x} d(m^2 + n^2).$$It is proved in the paper that, as $x o infty$,$$S(x) := A_1 x^2 log x + A_2 x^2 + O_epsilon (x^{frac32 +epsilon}),$$where $A_1$ and $A_2$ are certain constants and $epsilon$ is anyfixed positive real number.The result corrects a false formula given in a paper of Gafurovconcerning the same problem, and improves the error $O igl(x^{frac53} (log x)^9 igr)$ claimed by Gafurov.

【 授权许可】

Unknown