Mean values of arithmetic functions on a sparse set and applications
Volume 180 / 2026
Colloquium Mathematicum 180 (2026), 187-211
MSC: Primary 11A25; Secondary 11N05, 11L03
DOI: 10.4064/cm9568-10-2025
Published online: 6 May 2026
Abstract
Let $f$ be an arithmetic function satisfying some simple conditions. The aim of this paper is to establish some asymptotic estimates for the quantities $$ \psi _f(x) := \sum _{n\le x} \varLambda (n) f\Big(\Big[\frac{x}{n}\Big]\Big), \quad \,\, M\pi _f(x) := \sum _{p\le x} f\Big(\Big[\frac{x}{p}\Big]\Big) $$ for $x\to \infty $, where $\varLambda (n)$ is the von Mangoldt function and $[t]$ is the integer part of $t\in \mathbb {R}$. These generalise or sharpen some recent results of Saito–Suzuki–Takeda–Yoshida. As an application, we show that $$ \sum _{p\le x, \, [{x}/{p}]\, {\rm is}\,{\rm prime}} 1 \,\mathop{\sim }_{x\to \infty }\, \biggl(\sum_{p} \frac{1}{p(p+1)}\bigg)\frac{x}{\log x}\cdot $$
