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Source. Theorem 2.4, p. 5 of the author's manuscript, of Carl Pomerance, The first function and its iterates, in Connections in Discrete Mathematics, Cambridge University Press (2018), 125--138, as identified on the source card. Page numbers are those of the manuscript.

Statement

Here $s(n)=\sigma(n)-n$, $s_k$ is its $k$-th iterate, and $\beta$ is the Bosma–Kane constant of Theorem 1.1 (p. 2).

Theorem 2.4 (p. 5). Assume Conjecture 2.3. Then for each integer $k\ge2$ there is a set $A_k$ of asymptotic density 1 such that

$$\frac1x\sum_{\substack{n\le x\\ n\in A_k}}\log\bigl(s_k(n)/s_{k-1}(n)\bigr)\to\beta \qquad(x\to\infty).$$

The theorem is conditional, and the set $A_k$ depends on $k$. As printed, the sum runs over all $n\le x$ in $A_k$, not over even arguments, while $\beta$ was introduced on p. 2 as the limit of an average over even arguments $2n$. This page reports the statement as printed. After recording the Bosma–Kane asymptotic for the full sum $\sum_{1<n\le x}\log(s(n)/n)$, the paper says (p. 6) that it has no analogue of the theorem for the full sum of the terms $\log(s_k(n)/s_{k-1}(n))$, since that sum is presumably supported mainly on a set of $n$ of density 0.

Proof pointer

Proof on p. 6. Conjecture 2.3 gives, by induction, that $s_j^{-1}(B)$ has density 0 when $B$ has density 0. With $A$ the density-one set of $n$ with enough primes $p\parallel n$ in prescribed residue classes and $\omega(n)\le3\log_2n$, as in the proof of Proposition 2.1 (p. 4), $A_k$ is $A$ with the sets $s_j^{-1}(B)$, $j<k$, removed, where $B$ is the complement of $A$. For $n\in A_k$ every $s_j(n)$, $j<k$, lies in $A$, so the successive ratios $s_{j+1}(n)/s_j(n)$ are asymptotic to one another.

Dependencies

Conjecture 2.3 (assumed) and the argument of Proposition 2.1 (p. 4). Read depth: claims checked; the statement was read clause by clause on p. 5 and the proof for its structure only.

Bears on

• Problem 955: the theorem assumes the problem's statement (Conjecture 2.3) and derives a consequence from it; it gives no evidence for or against the problem.
• Problem 410: background only. The theorem concerns iterates of $s=\sigma-\mathrm{id}$, not of $\sigma$, and is conditional.