Logo
ScienceStack
Table of Contents
Fetching ...
Sign In

On almost primes in Piatetski-Shapiro sequences

Runbo Li

TL;DR

The paper proves that for 0.9985 < \\gamma < 1 there exist infinitely many primes p with \\Omega([p^{1/\\gamma}])\\le 5, improving earlier bounds. It employs Chen's switching principle and Iwaniec's linear sieve in a sieve framework, augmented by two new combinatorial lemmas to control error terms. The main technical achievement is a positive lower bound arising from a carefully balanced weighted sum W(A, x^{1/17.41}) and associated integral estimates, establishing the P5 bound for the almost-prime outputs. The work highlights a barrier to reaching P4 due to limitations in handling higher-dimensional exponential-sum error terms and Richert-type sieve weights, clarifying the method's current cap near P5 and its dependence on γ close to 1.

Abstract

The author proves that for $0.9985 < γ< 1$, there exist infinitely many primes $p$ such that $[p^{1/γ}]$ 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.

On almost primes in Piatetski-Shapiro sequences

TL;DR

The paper proves that for 0.9985 < \\gamma < 1 there exist infinitely many primes p with \\Omega([p^{1/\\gamma}])\\le 5, improving earlier bounds. It employs Chen's switching principle and Iwaniec's linear sieve in a sieve framework, augmented by two new combinatorial lemmas to control error terms. The main technical achievement is a positive lower bound arising from a carefully balanced weighted sum W(A, x^{1/17.41}) and associated integral estimates, establishing the P5 bound for the almost-prime outputs. The work highlights a barrier to reaching P4 due to limitations in handling higher-dimensional exponential-sum error terms and Richert-type sieve weights, clarifying the method's current cap near P5 and its dependence on γ close to 1.

Abstract

The author proves that for , there exist infinitely many primes such that 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.
Paper Structure (3 sections, 3 theorems, 76 equations)

This paper contains 3 sections, 3 theorems, 76 equations.

Table of Contents

  1. Introduction
  2. Two combinatorial lemmas
  3. Proof of Theorem 1.1

Key Result

Theorem 1.1

There are infinitely many $q \in \mathscr{P}_{\gamma}$ with $q = P_5$ for any $\gamma \in \left(0.9985, 1\right)$. Moreover, the following estimates holds for all sufficiently large $x$ provided that $\gamma \in \left(0.9985, 1\right)$.

Theorems & Definitions (5)

  • Theorem 1.1
  • Lemma 2.1
  • proof
  • Lemma 2.2
  • proof