PreprintarXiv licencearXiv
On a variant of the Piatetski-Shapiro prime number theorem
Abstract
Denote by Πγ(x) the number of primes p leqslant x such that [p1/γ] is also a prime. In this paper, it is proved that the expected asymptotic formula Πγ(x)=γx/((log x)2) (1+o(1)) 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.
Related
- Multiplicative recurrence of Piatetski--Shapiro and nil-Bohr sets
- Polynomial configurations and pointwise averages along Piatetski--Shapiro primes
- Polynomial configurations and pointwise averages along Piatetski-Shapiro sequences
- Linear equations in Piatetski-Shapiro primes
- Better than square-root cancellation and Gaussian behavior on Piatetski-Shapiro Sequences
BibTeX
@misc{long2026variant,
title = {{On a variant of the Piatetski-Shapiro prime number theorem}},
author = {Linji Long and Jinjiang Li and Min Zhang},
howpublished = {arXiv},
year = {2026},
month = oct,
doi = {10.48550/arxiv.2610.15140},
eprint = {2610.15140},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2610.15140},
}
RIS
TY - UNPB
TI - On a variant of the Piatetski-Shapiro prime number theorem
AU - Linji Long
AU - Jinjiang Li
AU - Min Zhang
PY - 2026
DA - 2026/10/10
PB - arXiv
DO - 10.48550/arxiv.2610.15140
UR - https://arxiv.org/abs/2610.15140
AB - 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 Π_γ(x)=γx/((log x)^2) (1+o(1)) 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.
ER -