On a variant of the prime number theorem

Abstract

In this paper, we can show that \begin{align*} S_{\Lambda}(x)=\sum_{1\leq n\leq x}\Lambda \left(\left[\frac{x}{n}\right]\right)= \sum_{n=1}^{\infty} \frac{\Lambda(n)}{n(n+1)}x +O\left(x^{7/15+1/195+\varepsilon}\right), \end{align*} where $\Lambda(n)$ is the von Mangdolt function. Moreover, we can also give a similar result related to the divisor function, which improves previous results.