Involves averaging arithmetic and integral partial functions over sparse set

2024/06/28 by Ye, Zhaoxi, Xu, Zhefeng
#FOS: Mathematics #Number Theory (math.NT)

paper · doi:10.48550/arxiv.2406.19788

Abstract

Let p be a prime number, k≥ 0 and f be a class of arithmetic functions satisfying some simple conditions. In this short paper, we study the asymptotical behaviour of summation function ψ_f,k(x):=∑_{n≤ x}Λ(n)\fracf ( [ (x)/(n) ] ) [ (x)/(n) ]^k ,~~~~~~~~~~~ π_f,k(x):=∑_{p≤ x}\fracf ( [ (x)/(p) ] ) [ (x)/(p) ]^k as x→ ∞ , where [ ⋅ ] is the integral part function, Λ(n) is the von Mangoldt function.

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