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DISTRIBUTION OF r-FREE INTEGERS OVER A FLOOR FUNCTION SET
Part of
- Multiplicative number theory
PINTHIRA TANGSUPPHATHAWAT, TEERAPAT SRICHAN, VICHIAN LAOHAKOSOL
Journal:

Bulletin of the Australian Mathematical Society / Volume 108 / Issue 1 / August 2023

Published online by Cambridge University Press:

23 November 2022, pp. 107-113

Print publication:

August 2023
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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.