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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.

##### MSC:
 11N37 Asymptotic results on arithmetic functions 11D45 Counting solutions of Diophantine equations 11D25 Cubic and quartic Diophantine equations
##### Keywords:
divisor sum; binary form; degree 4; quartic; factors
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##### References:
 [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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