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# On the Number of Divisors of the Quadratic Form $m^2+n^2$

Published:2000-06-01
Printed: Jun 2000
• Gang Yu
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## Abstract

For an integer $n$, let $d(n)$ denote the ordinary divisor function. This paper studies the asymptotic behavior of the sum $$S(x) := \sum_{m\leq x, n\leq x} d(m^2 + n^2).$$ It is proved in the paper that, as $x \to \infty$, $$S(x) := A_1 x^2 \log x + A_2 x^2 + O_\epsilon (x^{\frac32 + \epsilon}),$$ where $A_1$ and $A_2$ are certain constants and $\epsilon$ is any fixed positive real number. The result corrects a false formula given in a paper of Gafurov concerning the same problem, and improves the error $O \bigl( x^{\frac53} (\log x)^9 \bigr)$ claimed by Gafurov.
 Keywords: divisor, large sieve, exponential sums