On some sums involving small arithmetic functions

2026/07/20 by Zhai Wenguang
Mathematics · #math.NT

paper · pdf

Abstract

Let f be any arithmetic function and define S_f(x):=∑_{n≤ x}f([x/n]). If the function f is small, namely, f(n)≪ n^ε, then the error term E_f(x) in the asymptotic formula of S_f(x) has the form O(x^1/2+ε). In this paper, we shall study the mean square of E_f(x) and establish some new results of E_f(x) for some special functions.

Related

• The fractional sum of small arithmetic functions
• On a sum involving general arithmetic functions and the integral part function
• On error term estimates à la Walfisz for mean values of arithmetic functions
• Mean Values for a Class of Arithmetic Functions in Short Intervals
• Mean square of the error term in the asymmetric many dimensional divisor problem