On some problems of primes with the floor function

Abstract

Let $\left[x\right]$ be the largest integer not exceeding $x$. For $0<\theta \le 1$, let ${\pi }_{\theta }(x)$ denote the number of integers $n$ with $1\le n\le x^{\theta }$ such that $\left[\frac{x}{n}\right]$ is prime and $S_{\mathbb{P}}(x)$ denote the number of primes in the sequence ${\left\{\left[\frac{x}{n}\right]\right\}}_{n⩾1}$. 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 $$S_{\mathbb{P}}(x)=x\sum _p\frac{1}{p(p+1)}+O_{\epsilon }\left(x^{435/923+\epsilon }\right)$$ for $x\to \infty$. Thus improve the previous result due to Ma, Wu and the author.

8 pages. The second version is a complete improvement on the first version, so we only present the best results