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 $\epsilon >0$ the asymptotic formula $$\sum _{n\le x}\Lambda \left(\right[\frac{x}{n}\left]\right)=x\sum _{d\ge 1}\frac{\Lambda (d)}{d(d+1)}+O_{\epsilon }\left(x^{9/19+\epsilon }\right)(x\to \infty )$$ holds. This improves a recent result of Bordellès, which requires $\frac{97}{203}$ in place of $\frac{9}{19}$.

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