Polya-Radoux type results for some arithmetical functions

JOURNAL OF MATHEMATICAL ANALYSIS AND APPLICATIONS

卷:428

Article

关键词: Riemann integral;    Arithmetic functions;    Asymptotic results;    Polya theorem;    Radoux theorem;   

DOI  :  10.1016/j.jmaa.2015.03.062

来源: Elsevier

【 摘 要 】

Let h : [1, infinity) -> [0, infinity) be a function with the properties that there exists x(0) >= 1 such that h is an element of C-1 ([x(0), infinity)), h (x(0)) > 0, h' (x) > 0 for all x >= x(0) and lim(x ->infinity) h(x) = infinity. We prove that if a : N > [0, infinity) and k >= 1 then, Sigma(n <= x) a(n) similar to [h(x)](k) if and only if for every function f : [0,1] > R such that x(k-1) f(x) is Riemann integrable the following equality holds lim(x ->infinity) 1/[h(x)](k) Sigma(x0<= x) a(n) f(h(n)/h(x)) = k integral(1)(0) x(k-1) f(x) dx. Applications in the case of the prime numbers are given. (C) 2015 Elsevier Inc. All rights reserved.

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