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The asymptotic formula for the number of representations of an integer as a sum of five squares. (English) Zbl 0857.11019
Berndt, Bruce C. (ed.) et al., Analytic number theory. Vol. 1. Proceedings of a conference in honor of Heini Halberstam, May 16-20, 1995, Urbana, IL, USA. Boston, MA: Birkhäuser. Prog. Math. 138, 129-139 (1996).
The purpose of this article is to observe “that the asymptotic formula for the number of representations of an integer as a sum of five squares can be derived by a simple elementary argument from Jacobi’s formula for the number of representations of an integer as a sum of four squares. A by-product of ... [the] argument is an asymptotic formula for the sum \[ \sum_{|j |< \sqrt n} \sigma (n-j^2), \] where \(\sigma\) denotes the sum-of-divisors function.”
The Jacobi formula for four squares is \((*)\) \(r_4(n)=8 \sigma^*(n)\), \(n\in \mathbb{Z}^+\), where \(r_s(n)\) is the counting function for representations of \(n\) as a sum of \(s\) squares \((s\in \mathbb{Z}^+\) and \(\sigma^*(n) = \sigma(n)\), if \(4 \nmid n\); \(\sigma^*(n) = \sigma (n)-4 \sigma(n/4)\); if \(4\mid n\). The “simple elementary argument” starting from \((*)\) yields \((**)\) \(r_5(n)=\) singular series \(+O (n(1+\log n)^3)\), an improvement over the asymptotic formula one obtains for \(r_5(n)\) from the circle method. (Of course, as the author points out, Hardy used modular forms to show that the error term in \((**)\) is actually 0.)
The asymptotic formula for \(\sum_{|j |\leq \sqrt n} \sigma (n-j^2)\) given here involves the same error term as that in \((**)\). The paper closes with a conjectured exact formula for \(\sum_{|j |\leq \sqrt n} \sigma (n-j^2)\). The author and the reviewer have recently established this exact formula through the use of modular forms.
For the entire collection see [Zbl 0841.00014].

MSC:
11E25 Sums of squares and representations by other particular quadratic forms
11N37 Asymptotic results on arithmetic functions
11P55 Applications of the Hardy-Littlewood method
11P05 Waring’s problem and variants
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