Primes in the intersection of two Piatetski-Shapiro sets

Let $\pi (x;{\gamma }_1,{\gamma }_2)$ denote the number of primes $p$ with $p⩽x$ and $p=\lfloor n_1^{1\mathrm{/}{\gamma }_1}\rfloor =\lfloor n_2^{1\mathrm{/}{\gamma }_2}\rfloor$, where $\lfloor t\rfloor$ denotes the integer part of $t\in \mathbb{R}$ and $1\mathrm{/}2<{\gamma }_2<{\gamma }_1<1$ are fixed constants. In this paper, we show that $\pi (x;{\gamma }_1,{\gamma }_2)$ holds an asymptotic formula for $21\mathrm{/}11<{\gamma }_1+{\gamma }_2<2$, which constitutes an improvement upon the previous result of Baker [1].