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October, 2019 Waring-Goldbach Problem: Two Squares and Three Biquadrates
Yingchun Cai, Li Zhu
Taiwanese J. Math. 23(5): 1061-1071 (October, 2019). DOI: 10.11650/tjm/181107


Assume that $\psi$ is a function of positive variable $t$, monotonically increasing to infinity and $0 \lt \psi(t) \ll \log t/(\log \log t)$. Let $\mathcal{R}_{3}(n)$ denote the number of representations of the integer $n$ as sums of two squares and three biquadrates of primes and we write $\mathcal{E}_{3}(N)$ for the number of integers $n$ satisfying $n \leq N$, $n \equiv 5, 53, 101 \pmod{120}$ and \[ \left| \mathcal{R}_{3}(n) - \frac{\Gamma^{2}(1/2) \Gamma^{3}(1/4)}{\Gamma(7/4)} \frac{\mathfrak{S}_{3}(n) n^{3/4}}{\log^{5}n} \right| \geq \frac{n^{3/4}}{\psi(n) \log^{5}n}, \] where $0 \lt \mathfrak{S}_{3}(n) \ll 1$ is the singular series. In this paper, we prove \[ \mathcal{E}_{3}(N) \ll N^{23/48+\varepsilon} \psi^{2}(N) \] for any $\varepsilon \gt 0$. This result constitutes a refinement upon that of Friedlander and Wooley [2].


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Yingchun Cai. Li Zhu. "Waring-Goldbach Problem: Two Squares and Three Biquadrates." Taiwanese J. Math. 23 (5) 1061 - 1071, October, 2019.


Received: 28 March 2018; Revised: 7 August 2018; Accepted: 14 November 2018; Published: October, 2019
First available in Project Euclid: 21 November 2018

zbMATH: 07126938
MathSciNet: MR4012369
Digital Object Identifier: 10.11650/tjm/181107

Primary: 11N36, 11P32

Rights: Copyright © 2019 The Mathematical Society of the Republic of China


Vol.23 • No. 5 • October, 2019
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