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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