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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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