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Source. Lemma 2.1, p. 3, of Carl Pomerance, On amicable numbers, Analytic Number Theory: In Honor of Helmut Maier's 60th Birthday, Springer (2015), 321--327, doi:10.1007/978-3-319-22240-0_19, as identified on the source card. Labels and pages are those of the author's manuscript named there (pp. 1--7).

Statement

Notation (pp. 2--3). $P(n)$ is the largest prime factor of $n>1$, and $P(1)=1$. $\Sigma(x,y)$ is the number of squarefree $n\in[1,x]$ with $P(\sigma(n))\le y$.

Lemma 2.1 (p. 3, quoted). "For each fixed $\varepsilon>0$, we have $\Sigma(x,y)\le x\exp(-(1+o(1))u\log\log u)$ as $u\to\infty$, where $u=\log x/\log y$ and $(\log\log x)^{1+\varepsilon}\le y\le x$."

Proof pointer

No proof is written out. The paper states (p. 3) that the proof follows from small cosmetic changes to the proof of the same bound for $\Phi(x,y)$, the number of $n\in[1,x]$ with $P(\varphi(n))\le y$, in Banks, Friedlander, Pomerance and Shparlinski [2, Theorem 3.1] (p. 2): the factor $p+1$ in $\sigma(n)$ plays the role of $p-1$ in $\varphi(n)$, and the restriction to squarefree $n$ avoids the different treatment of higher prime powers. The paper notes (p. 2) that for $\Phi(x,y)$ the factor $\log\log u$, in place of de Bruijn's $\log u$ for $y$-smooth numbers, is heuristically expected to be correct, but no matching lower bound is known.

Dependencies

[2, Theorem 3.1] (Banks, Friedlander, Pomerance and Shparlinski, Fields Inst. Comm. 41 (2004)), not checked here. Read depth: claims checked; the statement was read clause by clause on p. 3; there is no proof in the paper to check.

Bears on

No problem page directly. The lemma is the input to step (vii) of the proof of Theorem 1.1 (p. 5), whose page states the relation to #830.