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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].

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

Citations:

Zbl 0898.11038
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