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Integers expressible in a given number of ways as a sum of two squares. (English) Zbl 0790.11032
Knopp, Marvin (ed.) et al., A tribute to Emil Grosswald: number theory and related analysis. Providence, RI: American Mathematical Society. Contemp. Math. 143, 37-45 (1993).
For positive integers \(\ell\) and \(m\), let \(S_{\ell,m}\) be the number of integers \(n\) such that there are exactly \(m\) solutions in integers \(u_ 1,\dots,u_ \ell\) of \(n= u^ 2_ 1+ u^ 2_ 2+\cdots+ u^ 2_ \ell\) with \(u_ 1\geq u_ 2\geq\cdots \geq u_ \ell\geq 0\). E. Grosswald [Enseign. Math., II. Sér. 30, 223-245 (1984; Zbl 0558.10039)] gave a complete description in most cases of the behavior of the counting function \(F_{\ell,m}(x)=\#\{n: n\in S_{\ell,m},\;n\leq x\}\). However, his results were incomplete when \(\ell=2\), in which case he conjectured that \[ F_{2,m}(x)\sim C_ m x(\log x)^{-1} (\log\log x)^ c, \] where \(2^ c\) is the highest power of 2 dividing \(m\) and \(C_ m\) is a positive constant depending only on \(m\).
In the present paper, the author proves this asymptotic equality by using a Tauberian theorem of H. Delange [Ann. Sci. Ec. Norm. Supér., III. Ser. 71, 213-242 (1954; Zbl 0056.331)].
For the entire collection see [Zbl 0773.00030].
11E25 Sums of squares and representations by other particular quadratic forms
40E05 Tauberian theorems, general
11M45 Tauberian theorems