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On the divisor-sum problem for binary forms. (English) Zbl 0913.11041
For some irreducible binary homogeneous polynomial \(f\) of degree \(k\geq 3\) with integer coefficients and some \(n\in\mathbb{Z}\), let \(r_f(n)\) denote the number of \((x_1,x_2)\in \mathbb{Z}^2\) with \(f(x_1,x_2)=n\). Let further \(d(n)\) denote the number of positive divisors of the non-zero integer \(n\). The author proves that for \(k=4\) \[ \sum_{0<| n|\leq N}d(n)r_f(n)= CN^{1/2}\log N+O(N^{1/2}\log\log N) \] holds with some constant \(C= C_f>0\). He also shows an analogous result for the case \(k=3\) that improves upon an estimate of G. Greaves [Acta Arith. 17, 1-28 (1970; Zbl 0198.37903)]. The proof is largely elementary and is based upon simple results from the geometry of numbers. The same method is applied to show that with some constant \(C'>0\) we have \[ \sum_{0<n\leq N}r_2(n)r_4(n)= C'N^{1/2}\log N+O(N^{1/2}\log\log N), \] where \(r_k(n)\) denotes the number of representations of \(n\) as the sum of two \(k\)th powers of integers.

MSC:
11N37 Asymptotic results on arithmetic functions
11E76 Forms of degree higher than two
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