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

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

We investigate the average order of the divisor function at values of binary cubic forms that are reducible over and discuss some 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, Tome 23 (2011) no. 3, pp. 579-602. doi : 10.5802/jtnb.778. http://www.numdam.org/articles/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. | MR | Zbl

[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. | MR | Zbl

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

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

[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. | MR | Zbl

[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. | Numdam | MR | Zbl

[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. | MR | Zbl

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

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