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A SHARP UPPER BOUND FOR THE SUM OF RECIPROCALS OF LEAST COMMON MULTIPLES II

Part of: Sequences and sets Elementary number theory

Published online by Cambridge University Press:  09 June 2022

SIAO HONG Open the ORCID record for SIAO HONG [Opens in a new window] ,
MEILING HUANG Open the ORCID record for MEILING HUANG [Opens in a new window]  and
YUANLIN LI Open the ORCID record for YUANLIN LI [Opens in a new window]
SIAO HONG*
Affiliation:
Department of Mathematics and Statistics, Brock University, St. Catharines, ON L2S 3A1, Canada
MEILING HUANG
Affiliation:
Department of Mathematics and Statistics, Brock University, St. Catharines, ON L2S 3A1, Canada e-mail: [email protected]
YUANLIN LI
Affiliation:
Department of Mathematics and Statistics, Brock University, St. Catharines, ON L2S 3A1, Canada e-mail: [email protected]
*
e-mail: [email protected]
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Abstract

Let n and k be positive integers with $n\ge k+1$ and let $\{a_i\}_{i=1}^n$ be a strictly increasing sequence of positive integers. Let $S_{n, k}:=\sum _{i=1}^{n-k} {1}/{\mathrm {lcm}(a_{i},a_{i+k})}$ . In 1978, Borwein [‘A sum of reciprocals of least common multiples’, Canad. Math. Bull. 20 (1978), 117–118] confirmed a conjecture of Erdős by showing that $S_{n,1}\le 1-{1}/{2^{n-1}}$ . Hong [‘A sharp upper bound for the sum of reciprocals of least common multiples’, Acta Math. Hungar. 160 (2020), 360–375] improved Borwein’s upper bound to $S_{n,1}\le {a_{1}}^{-1}(1-{1}/{2^{n-1}})$ and derived optimal upper bounds for $S_{n,2}$ and $S_{n,3}$ . In this paper, we present a sharp upper bound for $S_{n,4}$ and characterise the sequences $\{a_i\}_{i=1}^n$ for which the upper bound is attained.

Keywords

least common multiplereciprocalupper boundprime number theorem

MSC classification

Primary: 11A05: Multiplicative structure; Euclidean algorithm; greatest common divisors
Secondary: 11B65: Binomial coefficients; factorials; $q$-identities
Type
Research Article
Information
Bulletin of the Australian Mathematical Society , Volume 107 , Issue 1 , February 2023 , pp. 10 - 21
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Copyright
© The Author(s), 2022. Published by Cambridge University Press on behalf of Australian Mathematical Publishing Association Inc.

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Footnotes

This work was supported in part by research grants from the Natural Sciences and Engineering Research Council of Canada (Grant No. DDG-2019-04206 and RGPIN 2017-03903).

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