Geodesic
On the Number of Divisors of the Quadratic Form m2 + n2
Canadian mathematical bulletin, Tome 43 (2000) no. 2, pp. 239-256

Voir la notice de l'article provenant de la source Cambridge University Press

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.
DOI : 10.4153/CMB-2000-032-3
Mots-clés : 11G05, 14H52, Divisor, large sieve, exponential sums 
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/}
}
TY  - JOUR
AU  - Yu, Gang
TI  - On the Number of Divisors of the Quadratic Form m2 + n2
JO  - Canadian mathematical bulletin
PY  - 2000
SP  - 239
EP  - 256
VL  - 43
IS  - 2
UR  - http://geodesic.mathdoc.fr/articles/10.4153/CMB-2000-032-3/
DO  - 10.4153/CMB-2000-032-3
ID  - 10_4153_CMB_2000_032_3
ER  - 
%0 Journal Article
%A Yu, Gang
%T On the Number of Divisors of the Quadratic Form m2 + n2
%J Canadian mathematical bulletin
%D 2000
%P 239-256
%V 43
%N 2
%U http://geodesic.mathdoc.fr/articles/10.4153/CMB-2000-032-3/
%R 10.4153/CMB-2000-032-3
%F 10_4153_CMB_2000_032_3

[1] [1] Davenport, H., Multiplicative Number Theory. Graduate Texts in Math. 74, Revised by L, H.. Montgomery Springer, 1980. Google Scholar

[2] [2] Friedlander, J. B. and Iwaniec, H., The polynomial x2 + y4 captures its primes. preprint. Google Scholar

[3] [3] Fouvry, E. and Iwaniec, H., Gaussian primes. Acta Arith. 129(1997), 249–287. Google Scholar

[4] [4] Gafurov, N.,On the number of divisors of a quadratic form. Proc. Steklov Inst.Math. (1993), 137–148. Google Scholar

[5] [5] Hooley, C., On the number of divisors of quadratic polynomials. ActaMath. 110(1963), 97–114. Google Scholar

[6] [6] Iwaniec, H. and Mozzochi, C. J., On the divisor and circle problems. J.Number Theory 29(1988), 60–93. Google Scholar

[7] [7] Vaaler, J. D., Some extremal problems in Fourier analysis. Bull. Amer.Math. Soc. (2) 12(1985), 183–216. Google Scholar

Cité par Sources :