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 1204.11158Let \(F(x_1,x_2)\in\mathbb{Z}[x_1,x_2]\) 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.

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.

Reviewer: Roger Heath-Brown (Oxford)

##### MSC:

11N37 | Asymptotic results on arithmetic functions |

11D45 | Counting solutions of Diophantine equations |

11D25 | Cubic and quartic Diophantine equations |

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\textit{R. de la Bretèche} and \textit{T. D. Browning}, J. Reine Angew. Math. 646, 1--44 (2010; Zbl 1204.11158)

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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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