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

Autor:	Artūras Dubickas
Jazyk:	angličtina
Rok vydání:	2024
Předmět:	harmonic number fractional part arithmetic mean Euler’s constant Mathematics QA1-939
Zdroj:	Mathematics, Vol 12, Iss 23, p 3731 (2024)
Druh dokumentu:	article
ISSN:	2227-7390
DOI:	10.3390/math12233731
Popis:	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].
Databáze:	Directory of Open Access Journals
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