On a variant of the Piatetski-Shapiro prime number theorem

Source abstract

Denote by $Π_γ(x)$ the number of primes $p \leqslant x$ such that $[p^{1/γ}]$ is also a prime. In this paper, it is proved that the expected asymptotic formula $$\begin{equation*} Π_γ(x)=\frac{γx}{(\log x)^2} (1+o(1)) \end{equation*}$$ holds for $γ> 15/13$, which constitutes an improvement upon the previous result of Balog [2] who showed that the above asymptotic formula holds for $γ> 6/5$.