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On moments of gaps between consecutive square-free numbers

Tsz Ho Chan
Vol. 12 (2023), No. 4, 287–295
DOI: 10.2140/moscow.2023.12.287
Abstract

Let s1,s2,s3,… ⁡ be the set of square-free numbers in ascending order. We prove that the following asymptotic on moments of gaps between square-free numbers

∑ sk+1≤x(sk+1 − sk)γ ∼ B(γ)x, with some constant B(γ) > 0,

is true for 0 ≤ γ < 3.75. This improves the previous best range 0 ≤ γ < 3.6875.

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Keywords
square-free numbers, gaps, moments
Mathematical Subject Classification
Primary: 11N25
Milestones
Received: 15 April 2023
Revised: 28 September 2023
Accepted: 12 October 2023
Published: 8 December 2023
Authors
Tsz Ho Chan
Department of Mathematics
Kennesaw State University
Marietta, GA
United States
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