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Claim. J. Lambek and L. Moser, On integers $n$ relatively prime to $f(n)$, Canad. J. Math. 7 (1955), 155--158. Let $Q(x)$ count the $n\le x$ with $(n,f(n))=1$. Theorem 1: if $f$ is non-decreasing and $f^*(n)$, the number of $m$ with $f(m)=n$, is finite and non-decreasing, then

$$Q(x)=6\pi^{-2}x+O(f(x)\log f(x))+O(f^*(f(x))\log f(x))+O(xf(x)^{-1}).$$

Theorem 2 gives density $6\pi^{-2}$ when also $f(x)\log f(x)=o(x)$ and $f^*(f(x))\log f(x)=o(x)$. Example 1 and its extension: for $f(x)=\lfloor x^{1/k}\rfloor$ with $k>1$ an integer, $Q(x)=6\pi^{-2}x+O(x^{1-1/k}\log x)$.

Covers. The exponents $\alpha=1/k$ for integers $k\ge2$ of Problem 1149, not all $0<\alpha<1$. Bergelson and Richter credit the paper with every $0<\alpha<1$, but the paper states only $\alpha=1/k$. Its Theorem 2 does not reach other exponents as stated, because the number of $n$ with $\lfloor n^\alpha\rfloor=m$ need not be non-decreasing in $m$ (for $\alpha=2/3$ it is $2,3,2$ at $m=1,2,3$). The full statement is Delmer and Deshouillers's.

Depends on. No page of this wiki.

Acceptance. Refereed: Canadian Journal of Mathematics 7 (1955). The page name's date is the publication year; the day is unknown.