On the Range of Arithmetic Means of the Fractional Parts of Harmonic Numbers

In this paper, the limit points of the sequence of arithmetic means 1n∑m=1n{Hm}σ for n=1,2,3,… are studied, where Hm is the mth harmonic number with fractional part {Hm} and σ is a fixed positive constant. In particular, for σ=1, it is shown that the largest limit point of the above sequence is 1/(e−1)=0.581976…, its smallest limit point is 1−log(e−1)=0.458675…, and all limit points form a closed interval between these two constants. A similar result holds for the sequence 1n∑m=1nf({Hm}), n=1,2,3,…, where f(x)=xσ is replaced by an arbitrary absolutely continuous function f in [0,1].