Waring–Goldbach problem: two squares and three biquadrates

Abstract

Let $\mathcal{ℛ}\left(n\right)$ denote the number of representations of the positive integer $n$ as the sum of two squares and three biquadrates of primes and we write $\mathcal{ℰ}\left(N\right)$ for the number of positive integers $n$ satisfying $n\le N$, $n\equiv 5,53,101\left(mod120\right)$ and

$$\left|\mathcal{ℛ}\left(n\right)-\frac{{\mathrm{\Gamma }}^2\left(\frac{1}{2}\right){\mathrm{\Gamma }}^3\left(\frac{1}{4}\right)}{\mathrm{\Gamma }\left(\frac{7}{4}\right)}\frac{\mathfrak{𝔖}\left(n\right)n^{\frac{3}{4}}}{{log}^{5}n}\right|\ge \frac{n^{\frac{3}{4}}}{{log}^{\frac{11}{2}}n},$$

where $0<\mathfrak{𝔖}\left(n\right)\ll 1$ is the singular series. In this paper, we prove

$$\mathcal{ℰ}\left(N\right)\ll N^{\frac{15}{32}+𝜀}$$

for any $𝜀>0$. This result constitutes a refinement upon that of Friedlander and Wooley (2014).