Primes in the intersection of two Piatetski-Shapiro sets

Abstract

Let $π(x;γ_1,γ_2)$ denote the number of primes $p$ with $p\leqslant x$ and $p=\lfloor n^{1/γ_1}_1\rfloor=\lfloor n^{1/γ_2}_2\rfloor$, where $\lfloor t\rfloor$ denotes the integer part of $t\in\mathbb{R}$ and $1/2<γ_2<γ_1<1$ are fixed constants. In this paper, we show that $π(x;γ_1,γ_2)$ holds an asymptotic formula for $21/11<γ_1+γ_2<2$, which constitutes an improvement upon the previous result of Baker [1].