Kempner-like Harmonic Series

Inspired by a question asked on the list mathfun, we revisit Kempner-like series, i.e., harmonic sums...1/n where the integers n in the summation have "restricted" digits. First we give a short proof that limk→∞(... = 2 log 2, where s2(n) is the sum of the binary digits of the integer n. T...

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Bibliographic Details
Published inThe American mathematical monthly Vol. 131; no. 9; pp. 775 - 783
Main Authors Allouche, Jean-Paul, Morin, Claude
Format Journal Article
LanguageEnglish
Published Washington Taylor & Francis Ltd 20.10.2024
Subjects
Binary digits
Harmonic analysis
Mathematical problems
Short stories
Theorems
British Columbia Canada
Canada
Online AccessGet full text
ISSN0002-9890
1930-0972
DOI10.1080/00029890.2024.2380232

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Summary:Inspired by a question asked on the list mathfun, we revisit Kempner-like series, i.e., harmonic sums...1/n where the integers n in the summation have "restricted" digits. First we give a short proof that limk→∞(... = 2 log 2, where s2(n) is the sum of the binary digits of the integer n. Then we give a generalization that addresses the case where s2(n) is replaced with sb(n), the sum of b-ary digits in base b: we prove that limk→∞... = (2 log b)/(b − 1). Finally we indicate that other generalizations could be studied: the sum of digits in base 2 could be replaced with, e.g., the function a11(n) of -possibly overlapping- 11 in the base-2 expansion of n, for which one can obtain limk→∞... 1/n = 4 log 2. (ProQuest: ... denotes formulae omited.)
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ISSN:0002-9890
1930-0972
DOI:10.1080/00029890.2024.2380232