Square-Free Harmonic Numbers: Basic Properties

Abstract

We study a variant of harmonic numbers defined by restricting the summation to square?free indices, denoted by Hₙˢq. Using the Mobius function to detect square?free integers, we establish a relation between Hₙˢq and the ordinary harmonic numbers Hₙ. We derive an asymptotic expansion for Hₙˢq with an explicit constant involving the Riemann zeta function. Moreover, we obtain the ordinary generating function, a convolution identity, and the arithmetic mean of these numbers.