MEAN VALUES OF ARITHMETIC FUNCTIONS ON A SPARSE SET AND APPLICATIONS
Résumé
Let $f$ be an arithmetic function satisfying some simple conditions.
The aim of this paper is to establish some asymptotic estimates for quantities
$$
\psi_f(x) := \sum_{n\le x} \Lambda(n) f\Big(\Big[\frac{x}{n}\Big]\Big),
\qquad
\pi_f(x) := \sum_{p\le x} f\Big(\Big[\frac{x}{p}\Big]\Big)
$$
for $x\to\infty$,
where $\Lambda(n)$ is the von Mangoldt function and $[t]$ is the integral part
of $t\in \R$.
These generalise or sharpen some recent results of Saito-Suzuki-Takeda-Yoshida.
As an application, we show that
$$
\sum_{p\le x, \, [\frac{x}{p}]\;\text{\rm is prime}} 1
\;\mathop{\sim}_{x\to\infty}\; \Big(\sum_{p} \frac{1}{p(p+1)}\Big)\frac{x}{\log x}\cdot
$$
Origine : Fichiers produits par l'(les) auteur(s)