On the Special Value Distribution of Two Classes of Small Multiplicative Functions

Abstract

For any real number x, [x] denotes the integer part of x. \(\cal{F}\)₁, \(\cal{F}\)₂ denote two multiplicative function classes which are small in numerical sense. In this paper, we study the summation \(\sum\nolimits_{{n\leq x}}f([x/n])\) for f ∈ \(\cal{F}\)₁. As specific cases, we take d^(e)(n), β(n), a(n), μ₂(n) denoting the number of exponential divisors of n, the number of square-full divisors of n, the number of non-isomorphic Abelian groups of order n, and the characteristic function of the square-free integers, respectively. In the case of μ₂(n), we improved the result of Liu, Wu and Yang. The sums shaped like \(\sum\nolimits_{{n\leq x}}f([x/n]+f([x/n]))\) for f ∈ \(\cal{F}\)₂ are also researched.