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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 divisor, large sieve, exponential sums
MSC Classifications: 11G05, 14H52 show english descriptions Elliptic curves over global fields [See also 14H52]
Elliptic curves [See also 11G05, 11G07, 14Kxx]
11G05 - Elliptic curves over global fields [See also 14H52]
14H52 - Elliptic curves [See also 11G05, 11G07, 14Kxx]
 

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