On a sum involving the Mangoldt function

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 the asymptotic formula

$$\begin{aligned} \sum _{n\leqslant x} \Lambda \Big (\Big [\frac{x}{n}\Big ]\Big ) = x\sum _{d\geqslant 1}\frac{\Lambda (d)}{d(d+1)} + O_{\varepsilon }\big (x^{35/71+\varepsilon }\big ) \end{aligned}$$

holds as $x\rightarrow \infty$ for any $\varepsilon >0$.