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DISTRIBUTION OF r-FREE INTEGERS OVER A FLOOR FUNCTION SET

Part of: Multiplicative number theory

Published online by Cambridge University Press:  23 November 2022

PINTHIRA TANGSUPPHATHAWAT Open the ORCID record for PINTHIRA TANGSUPPHATHAWAT [Opens in a new window] ,
TEERAPAT SRICHAN Open the ORCID record for TEERAPAT SRICHAN [Opens in a new window]  and
VICHIAN LAOHAKOSOL Open the ORCID record for VICHIAN LAOHAKOSOL [Opens in a new window]
PINTHIRA TANGSUPPHATHAWAT
Affiliation:
Department of Mathematics, Faculty of Science and Technology, Phranakhon Rajabhat University, Bangkok 10220, Thailand e-mail: t.pinthira@hotmail.com
TEERAPAT SRICHAN*
Affiliation:
Department of Mathematics, Faculty of Science, Kasetsart University, Bangkok 10900, Thailand
VICHIAN LAOHAKOSOL
Affiliation:
Department of Mathematics, Faculty of Science, Kasetsart University, Bangkok 10900, Thailand e-mail: fscivil@ku.ac.th
*
e-mail: fscitrp@ku.ac.th
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Abstract

For a positive integer $r\geq 2$, a natural number n is r-free if there is no prime p such that $p^r\mid n$. Asymptotic formulae for the distribution of r-free integers in the floor function set $S(x):=\{\lfloor x/ n \rfloor :1\leq n\leq x\}$ are derived. The first formula uses an estimate for elements of $S(x)$ belonging to arithmetic progressions. The other, more refined, formula makes use of an exponent pair and the Riemann hypothesis.

Keywords

floor functionr-free integer

MSC classification

Primary: 11N64: Other results on the distribution of values or the characterization of arithmetic functions
Secondary: 11N69: Distribution of integers in special residue classes
Type
Research Article
Information
Bulletin of the Australian Mathematical Society , Volume 108 , Issue 1 , August 2023 , pp. 107 - 113
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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 financially supported by the Office of the Permanent Secretary, Ministry of Higher Education, Science, Research and Innovation, Grant No. RGNS 63-40.

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