On a variant of the Piatetski-Shapiro prime number theorem

Linji Long; Jinjiang Li; Min Zhang

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arXiv · 2610.15140

On a variant of the Piatetski-Shapiro prime number theorem

Linji Long·Jinjiang Li·Min Zhang

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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 $γ> 6/5$.

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BibTeX RIS

Linji Long, Jinjiang Li, Min Zhang. 2026-10-10. On a variant of the Piatetski-Shapiro prime number theorem. https://arxiv.org/abs/2610.15140

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