Mean values of arithmetic functions on a sparse set and applications

Let f be an arithmetic function satisfying some simple conditions. The aim of this paper is to establish some asymptotic estimates for the quantities ψf(x):=∑n≤xΛ(n)f([xn]),Mπf(x):=∑p≤xf([xp]) for x→∞, where Λ(n) is the von Mangoldt function and [t] is the integer part of t∈R. These generalise or sharpen some recent results of Saito–Suzuki–Takeda–Yoshida. As an application, we show that ∑p≤x,[x/p]isprime1∼x→∞(∑p1p(p+1))xlogx⋅