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On the divisor function problem for binary forms of degree 4. (Le problème des diviseurs pour des formes binaires de degré 4.) (French. English summary) Zbl 1204.11158
Let \(F(x_1,x_2)\in\mathbb{Z}[x_1,x_2]\) be a form of degree 4, which factors over \(\mathbb{Z}\) as a product two of linearly independent linear forms and an irreducible quadratic form. The goal of the present paper is to estimate the divisor sum \[ T(X)=\sum_{x_1,x_2}d(F(x_1,x_2)) \] where the summation runs over vectors \((x_1,x_2)\in\mathbb{Z}^2\cap X\mathcal{R}\), for some fixed bounded convex region \(\mathcal{R}\). Forms of degree up to 3 are relatively easy, see G. Greaves [Acta Arith. 17, 1–28 (1970; Zbl 0198.37903)].
Irreducible quartic forms were handled by S. Daniel [J. Reine Angew. Math. 507, 107–129 (1999; Zbl 0913.11041)], and a related sum with four linear factors occurs in the reviewer’s work [Number theory and algebraic geometry. Lond. Math. Soc. Lect. Note Ser. 303, 133–176 (2003; Zbl 1161.11387)]. The authors explain that their methods should enable them to handle all reducible quartic forms.
Under mild supplementary conditions it is shown that \[ T(X)=CX^2(\log X)^3+O(X^2(\log X)^{2+\varepsilon}) \] for any fixed \(\varepsilon>0\). The constant \(C\) is given explicitly, and has a natural adelic interpretation. The basic idea behind the proof is to reduce to counting points in \(X\mathcal{R}\) which lie in various lattices. A result of G. Marasingha [Acta Arith. 124, No. 4, 327–355 (2006; Zbl 1146.11047)] is used for this purpose. However there are a number of complications of detail to be overcome in computing the number of lattices which arise.

11N37 Asymptotic results on arithmetic functions
11D45 Counting solutions of Diophantine equations
11D25 Cubic and quartic Diophantine equations
Full Text: DOI
[1] DOI: 10.4064/aa125-3-6 · Zbl 1159.11035
[2] DOI: 10.1112/S0010437X08003692 · Zbl 1234.11132
[3] Daniel S., Math. 507 pp 107– (1999)
[4] DOI: 10.4064/aa124-4-3 · Zbl 1146.11047
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