On some sums involving the integral part function

TL;DR

This paper investigates sums involving the integral part function and various number-theoretic functions, providing improved bounds and asymptotic formulas that enhance previous results in the field.

Contribution

It establishes new asymptotic estimates for sums of number-theoretic functions involving the integral part, improving upon earlier bounds by Bordellès.

Findings

Derived asymptotic formulas with explicit error terms

Improved bounds for sums involving divisor and prime factor functions

Enhanced understanding of the distribution of square-free and divisor-related functions

Abstract

Denote by $\tau$ k (n), $\omega$(n) and $\mu$ 2 (n) the number of representations of n as product of k natural numbers, the number of distinct prime factors of n and the characteristic function of the square-free integers, respectively. Let [t] be the integral part of real number t. For f = $\omega$, 2 $\omega$ , $\mu$ 2 , $\tau$ k , we prove that n x f x n = x d 1 f (d) d(d + 1) + O $\epsilon$ (x $\theta$ f +$\epsilon$) for x $\rightarrow$ $\infty$, where $\theta$ $\omega$ = 53 110 , $\theta$ 2 $\omega$ = 9 19 , $\theta$ $\mu$2 = 2 5 , $\theta$ $\tau$ k = 5k--1 10k--1 and $\epsilon$ > 0 is an arbitrarily small positive number. These improve the corresponding results of Bordell{\`e}s.