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

Abstract

Let $P_r$ denote an integer with at most $r$ prime factors counted with multiplicity. In this paper we prove that for some $\lambda <\frac{1}{12}$, the inequality $\{\sqrt{p}\}<p^{-\lambda }$ 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 $\lambda =\frac{1}{15.1}$, which improves Cai's $\frac{1}{15.5}$.

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