A remark on the distribution of $\sqrt{p}$ modulo one involving primes of special type II

TL;DR

This work studies the inequality $\{\sqrt{p}\}<p^{-\lambda}$ for primes $p$ with $p+2=P_r$ (where $P_r$ is an $r$-almost-prime) and proves that for $r=4,5,6,7$ there exist infinitely many such primes with explicit $\lambda$ values, notably $\lambda=1/15.1$ when $r=4$. The authors combine a refined Rosser-type sieve, a mean-value theorem due to Cai et al., and Chen's switching to handle the mixed problem of almost-prime constraints and a Diophantine condition on $\sqrt{p}$, with Buchstab-type integral analysis governing the main terms. The core is to bound the count $S_r$ of prime solutions by $S_{r,1}-S_{r,2}$, where $S_{r,1}$ is estimated from below and $S_{r,2}$ from above; after introducing auxiliary sets $\mathcal{B}^1,\mathcal{B}^2$ and evaluating associated integrals $T_r$, the authors reduce the problem to explicit positivity checks that are verified numerically for the claimed $\lambda$ values. These results extend previous bounds (e.g., Cai's $1/15.5$) and demonstrate the effectiveness of the combined sieve-mean-value-Chen framework in mixed-prime problems.

Abstract

Let $P_{r}$ denote an integer with at most $r$ prime factors counted with multiplicity. In this paper we prove that for some $λ< \frac{1}{12}$, the inequality $\{\sqrt{p}\}<p^{-λ}$ has infinitely many solutions in primes $p$ such that $p+2=P_r$, where $r= 4, 5, 6, 7$. Specially, when $r = 4$ we obtain $λ= \frac{1}{15.1}$, which improves Cai's $\frac{1}{15.5}$.

Theorems & Definitions (9)

• Definition 1.1
• Theorem 1.2
• Theorem 1.3
• Theorem 1.4
• Lemma 2.1
• Lemma 2.2
• Lemma 2.3
• Lemma 2.4
• proof