Primes in the intersection of two Piatetski-Shapiro sets

TL;DR

This paper establishes an asymptotic formula for the count of primes that are simultaneously in two Piatetski-Shapiro sets, improving previous results by extending the range of parameters for which the formula holds.

Contribution

It proves an asymptotic formula for primes in the intersection of two Piatetski-Shapiro sets for a broader parameter range than previously known.

Findings

Asymptotic formula for $\pi(x;\gamma_1,\gamma_2)$ established

Range of $\gamma_1+\gamma_2$ extended to $21/11<\gamma_1+\gamma_2<2$

Improves upon Baker's earlier results

Abstract

Let $\pi(x;\gamma_1,\gamma_2)$ denote the number of primes $p$ with $p\leqslant x$ and $p=\lfloor n^{1/\gamma_1}_1\rfloor=\lfloor n^{1/\gamma_2}_2\rfloor$, where $\lfloor t\rfloor$ denotes the integer part of $t\in\mathbb{R}$ and $1/2<\gamma_2<\gamma_1<1$ are fixed constants. In this paper, we show that $\pi(x;\gamma_1,\gamma_2)$ holds an asymptotic formula for $21/11<\gamma_1+\gamma_2<2$, which constitutes an improvement upon the previous result of Baker [1].