On some problems of primes with the floor function

Abstract

Let $\left[x\right]$ be the largest integer not exceeding $x$. For $0<\theta \leq 1$, let $\pi_{\theta}(x)$ denote the number of integers $n$ with $1 \leq n \leq x^{\theta}$ such that $\left[\frac{x}{n}\right]$ is prime and $S_{\mathbb{P}}(x)$ denote the number of primes in the sequence $\left\{\left[\frac{x}{n}\right]\right\}_{n \geqslant 1}$. In this paper, we obtain the asymptotic formula $$ \pi_{\theta}(x)=\frac{x^{\theta}}{(1-\theta) \log x}+O\left(x^{\theta}(\log x)^{-2}\right) $$ provide that $\frac{435}{923}<\theta<1$, and prove that $$ S_{\mathbb{P}}(x)=x\sum_{p} \frac{1}{p(p+1)}+O_{\varepsilon}\left(x^{435/923+\varepsilon}\right) $$ for $x \rightarrow \infty$. Thus improve the previous result due to Ma, Wu and the author.