On a variant of the Piatetski-Shapiro prime number theorem

Denote by ${\mathrm{\Pi }}_{\gamma }(x)$ the number of primes $p⩽x$ such that $[p^{1\mathrm{/}\gamma }]$ 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 $\gamma >15\mathrm{/}13$, which constitutes an improvement upon the previous result of Balog [2] who showed that the above asymptotic formula holds for $\gamma >6\mathrm{/}5$.