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

Artūras Dubickas ()
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Artūras Dubickas: Institute of Mathematics, Faculty of Mathematics and Informatics, Vilnius University, Naugarduko 24, LT-03225 Vilnius, Lithuania

Mathematics, 2024, vol. 12, issue 23, 1-11

Abstract: In this paper, the limit points of the sequence of arithmetic means 1 n ∑ m = 1 n { H m } σ for n = 1 , 2 , 3 , … are studied, where H m is the m th harmonic number with fractional part { H m } 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 1 n ∑ m = 1 n f ( { H m } ) , n = 1 , 2 , 3 , … , where f ( x ) = x σ is replaced by an arbitrary absolutely continuous function f in [ 0 , 1 ] .

Keywords: harmonic number; fractional part; arithmetic mean; Euler’s constant (search for similar items in EconPapers)
JEL-codes: C (search for similar items in EconPapers)
Date: 2024
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