On fractional sum of the von Mangoldt function
Colloquium Mathematicum
MSC: Primary 11N37; Secondary 11L07
DOI: 10.4064/cm9418-10-2024
Published online: 2 December 2024
Abstract
Let $\Lambda (n)$ be the von Mangoldt function, and let $[t]$ be the integral part of real number $t$. We prove the asymptotic formula \[ \sum _{n\le x}\Lambda \left(\left[ \frac{x}{n}\right]\right) =x\sum _{d=1}^\infty \frac{\Lambda (d)}{d(d+1)}+O( x^{22/47+\varepsilon })\quad\ \text{for any $\varepsilon \gt 0$.} \]
