On the Number of Divisors of the Quadratic Form m2 + n2
On the Number of Divisors of the Quadratic Form m2 + n2
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DOI：
10.4153/cmb-2000-032-3
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发表时间：
2000-06
期刊：
影响因子：
--
通讯作者：
Gang Yu
中科院分区：
文献类型：
--
作者：
Gang Yu
Abstract 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.
Abstract 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.