DISTRIBUTION OFr-FREE INTEGERS OVER A FLOOR FUNCTION SET

Abstract

AbstractFor a positive integer$r\geq 2$, a natural numbernisr-free if there is no primepsuch that$p^r\mid n$. Asymptotic formulae for the distribution ofr-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.