On almost primes in Piatetski-Shapiro sequences

TL;DR

This paper proves the existence of infinitely many primes p where the integer part of p^{1/γ} has at most 5 prime factors for γ between 0.9985 and 1, improving previous results.

Contribution

It advances the understanding of prime distributions in Piatetski-Shapiro sequences by reducing the number of prime factors in the sequence elements.

Findings

Infinitely many primes p with [p^{1/γ}] having ≤5 prime factors for 0.9985<γ<1

Improves previous bounds on prime factors in Piatetski-Shapiro sequences

Extends the range of γ for which such primes are known to exist

Abstract

The author proves that for $0.9985 < \gamma < 1$, there exist infinitely many primes $p$ such that $[p^{1/\gamma}]$ has at most 5 prime factors counted with multiplicity. This gives an improvement upon the previous results of Banks-Guo-Shparlinski and Xue-Li-Zhang.