On some problems of primes with the floor function

TL;DR

This paper investigates the distribution of primes related to the floor function, deriving asymptotic formulas for certain prime-counting functions and improving previous results in the field.

Contribution

It provides new asymptotic formulas for prime counts involving the floor function and refines earlier bounds for these functions.

Findings

Derived an asymptotic formula for rac{ heta}{x} with specified rac{435}{923}<rac{ heta}{1}

Established an asymptotic for the sum over primes in a specific sequence

Improved previous bounds on prime-related functions involving 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.