A variant of the prime number theorem

Abstract

Let $\Lambda(n)$ be the von Mangoldt function, and let $[t]$ be the integral part of real number $t$. In this note, we prove that for any $\varepsilon>0$ the asymptotic formula $$ \sum_{n\le x} \Lambda\Big(\Big[\frac{x}{n}\Big]\Big) = x\sum_{d\ge 1} \frac{\Lambda(d)}{d(d+1)} + O_{\varepsilon}\big(x^{9/19+\varepsilon}\big) \qquad (x\to\infty)$$ holds. This improves a recent result of Bordell\`es, which requires $\frac{97}{203}$ in place of $\frac{9}{19}$.