On the quadratic Waring-Goldbach problem with primes in Piatetski-Shapiro sets

Abstract

In this paper, it is proved that, for any $\gamma_1,\gamma_2,\gamma_3,\gamma_4,\gamma_5\in(\frac{28}{29},1)$, every sufficiently large integer $n$ subject to $n\equiv5\pmod{24}$ can be represented as the sum of five squares of primes, i.e., \begin{equation*} n=p_1^2+p_2^2+p_3^2+p_4^2+p_5^2, \end{equation*} such that $p_i=\lfloor m_i^{1/\gamma_i}\rfloor$ for some $m_i\in\mathbb{N}^+$ for each $1\leqslant i\leqslant 5$. This result constitutes an improvement upon the previous result of Zhang and Zhai [29].