Mikawa, Hiroshi On the sum of three squares of primes. (English) Zbl 0906.11052 Motohashi, Y. (ed.), Analytic number theory. Proceedings of the 39th Taniguchi international symposium on mathematics, Kyoto, Japan, May 13–17, 1996. Cambridge: Cambridge University Press. Lond. Math. Soc. Lect. Note Ser. 247, 253-264 (1997). For any positive integer \(n\), let \(R(n)\) denote the weighted number of representations of \(n\) in the form \(p_1^2+ p_2^2+ p_3^2= n\), where \(p_j\) are primes; i.e., \[ R(n)= \int_0^1 \biggl( \sum_{p^2\leq 2x} e(\alpha p^2)\log p\biggr)^3 e(-n\alpha) d\alpha, \] where \(e(t)= \exp(i2\pi t)\) for any real \(t\). For any positive integer \(q\), let \(r(q,n)\) denote the number of solutions of the congruence equation \(n_1^2+ n_2^2+ n_3^2\equiv n\pmod q\), where \(n_j\) are integers satisfying \(\gcd (n_j,q)=1\). Define \(\mathbb{H}= \{n: n\equiv 3\pmod {24}\), \(5\nmid n\}\) and \[ {\mathfrak S}(n)= \frac{8r(8,n)} {\phi(8)^3} \prod_{p\geq 3} \frac{pr(p,n)} {\phi(p)^3}, \] where \(\phi(m)\) denotes the Euler function and \(p\) is prime. The author proves: Let \(\theta>1/2\) and \(A>0\) be given. Then for \(x>0\) we have \[ \sum_{\substack{ x<n\leq x+x^\theta\\ n\in\mathbb{H}}} | R(n)- {\mathfrak S}(n) \pi/\sqrt{n}/4|^2\ll x^{\theta+1}(\log x)^{-A}, \] where \(\ll\) is the Vinogradov symbol. The result sharpens the previous \(\theta> 3/4\) by J. Liu and T. Zhan [Arch. Math. 69, 375-390 (1997; Zbl 0898.11038)].For the entire collection see [Zbl 0874.00035]. Reviewer: M.-C.Liu (Hongkong) Cited in 1 ReviewCited in 3 Documents MSC: 11P32 Goldbach-type theorems; other additive questions involving primes 11P05 Waring’s problem and variants 11P55 Applications of the Hardy-Littlewood method 11D85 Representation problems Keywords:sum of three squares of primes; Waring-Goldbach problem; Hardy-Littlewood method; weighted number of representations; congruence equation Citations:Zbl 0898.11038 PDFBibTeX XMLCite \textit{H. Mikawa}, Lond. Math. Soc. Lect. Note Ser. 247, 253--264 (1997; Zbl 0906.11052)