On the Number of Divisors of the Quadratic Form m2 + n2
Canadian mathematical bulletin, Tome 43 (2000) no. 2, pp. 239-256
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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.
Yu, Gang. On the Number of Divisors of the Quadratic Form m2 + n2. Canadian mathematical bulletin, Tome 43 (2000) no. 2, pp. 239-256. doi: 10.4153/CMB-2000-032-3
@article{10_4153_CMB_2000_032_3,
author = {Yu, Gang},
title = {On the {Number} of {Divisors} of the {Quadratic} {Form} m2 + n2},
journal = {Canadian mathematical bulletin},
pages = {239--256},
year = {2000},
volume = {43},
number = {2},
doi = {10.4153/CMB-2000-032-3},
url = {http://geodesic.mathdoc.fr/articles/10.4153/CMB-2000-032-3/}
}
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