The divisor problem for binary cubic forms
Journal de théorie des nombres de Bordeaux, Volume 23 (2011) no. 3, pp. 579-602

We investigate the average order of the divisor function at values of binary cubic forms that are reducible over ℚ and discuss some applications.

Nous étudions l’ordre moyen du nombre de diviseurs des valeurs de certaines formes binaires cubiques qui ne sont pas irréductibles sur ℚ et discutons quelques applications.

DOI: 10.5802/jtnb.778
Classification: 11N37, 11D25

Browning, Tim  1

1 School of Mathematics University of Bristol Bristol BS8 1TW United Kingdom
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Browning, Tim. The divisor problem for binary cubic forms. Journal de théorie des nombres de Bordeaux, Volume 23 (2011) no. 3, pp. 579-602. doi: 10.5802/jtnb.778

[1] V. V. Batyrev and Yu. I. Manin, Sur le nombre des points rationnels de hauteur borné des variétés algébriques. Math. Ann. 286 (1990), 27–43. | Zbl | MR

[2] R. de la Bretèche and T. D. Browning, Binary linear forms as sums of two squares. Compositio Math. 144 (2008), 1375–1402. | MR

[3] R. de la Bretèche and T. D. Browning, Le problème des diviseurs pour des formes binaires de degré 4. J. reine angew. Math. 646 (2010), 1–44. | Zbl | MR

[4] S. Daniel, On the divisor-sum problem for binary forms. J. reine angew. Math. 507 (1999), 107–129. | Zbl | MR

[5] W. Duke, J. B. Friedlander and H. Iwaniec, A quadratic divisor problem. Inventiones Math. 115 (1994), 209–217. | Zbl | MR

[6] T. Estermann, Über die Darstellung einer Zahl als Differenz von swei Produkten. J. reine angew. Math. 164 (1931), 173–182. | Zbl

[7] G. Greaves, On the divisor-sum problem for binary cubic forms. Acta Arith. 17 (1970), 1–28. | Zbl | MR

[8] A. E. Ingham, Some asymptotic formulae in the theory of numbers. J. London Math. Soc. 2 (1927), 202–208.

[9] Y. Motohashi, The binary additive divisor problem. Ann. Sci. École Norm. Sup. 27 (1994), 529–572. | Zbl | MR | Numdam

[10] M. Robbiani, On the number of rational points of bounded height on smooth bilinear hypersurfaces in biprojective space. J. London Math. Soc. 63 (2001), 33–51. | Zbl | MR

[11] C. V. Spencer, The Manin Conjecture for x 0 y 0 +⋯+x s y s =0. J. Number Theory 129 (2009), 1505–1521. | Zbl | MR

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