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Source. The Theorem of Section 2, p. 184, of Carl Pomerance, On the distribution of amicable numbers. II, J. Reine Angew. Math. 325 (1981), 183--188, doi:10.1515/crll.1981.325.183, as identified on the source card. The paper also displays the bound as (2) on p. 183.

Statement

Setting (p. 183). Let $\sigma(n)$ be the sum of the divisors of $n$ and $s(n)=\sigma(n)-n$. Natural numbers $n,m$ form an amicable pair when $s(n)=m$ and $s(m)=n$; $n$ is an amicable number when it belongs to an amicable pair, equivalently when $s(s(n))=n$. The definition does not require $n\ne m$, so perfect numbers are amicable numbers here. $A(x)$ is the number of amicable numbers not exceeding $x$.

Theorem (p. 184, quoted). "For all large $x$, $A(x)\leqq x/e^{(\log x)^{1/3}}$."

Consequences stated in Section 1 (p. 183). The paper notes that the bound implies at once that the sum of the reciprocals of the amicable numbers is finite, which it says was not known before, and that it settles Erdős's conjecture (P. Erdős, On amicable numbers, Publ. Math. Debrecen 4 (1955), 108--111) that $A(x)=O(x/(\log x)^k)$ for every $k$. The previous best bound, from Pomerance's earlier paper (J. Reine Angew. Math. 293/294 (1977), 217--222), was $A(x)\le x\exp\{-c(\log\log\log x\,\log\log\log\log x)^{1/2}\}$ for all large $x$ with some positive constant $c$.

Remark (p. 187). The paper states without proof that small alterations of the argument give some $c>0$ with

$$A(x)\ll x\exp\{-c(\log x\,\log\log x)^{1/3}\}.$$

Read depth. Claims checked: the setting, the Theorem, the consequences and the Remark were read clause by clause on the printed pages. The proof (pp. 184--187) was read for its structure only, not checked step by step.

Proof pointer

Section 2, pp. 184--187. With $l=e^{(\log x)^{1/3}}$ and $L=e^{\frac18(\log x)^{2/3}\log\log x}$, and using $s(n)\le2x\log\log x$ for large $x$ and $n\le x$, the proof discards $o(x/l)$ amicable $n\le x$ at each of the following steps, writing $P(n)$ for the largest prime factor of $n$: (i) $P(n)$ and $P(s(n))$ are at least $L^2$ (by de Bruijn's count of smooth numbers); (ii) no $k^a$ with $a\ge2$ and $k^a\ge l^3$ divides $n$ or $s(n)$; (iii) every prime dividing both $n$ and $\sigma(n)$ is below $l^4$; (iv) $n/P(n)$ and $s(n)/P(s(n))$ are at least $L$, since the cofactors $m$, $m'$ determine $n$; (v) $P(\sigma(m))$ and $P(\sigma(m'))$ are at least $l^4$ for those cofactors, via a count of factorizations following Canfield, Erdős and Pomerance with the parameter $1-(\log x)^{-1/3}$ (inequality (6), p. 186). For the $n$ that remain, a large prime $r$ dividing $\sigma(m)$ forces primes $q\Vert m$ and $q'\Vert s(n)$ with $q\equiv q'\equiv-1\pmod r$, which fixes $P(n)$ in one residue class modulo $q'$; summing over $r$, $q$, $m$ and $q'$ gives $o(x/l)$ (p. 187).

Dependencies

N. G. de Bruijn, On the number of integers $\le x$ and free of prime factors $>y$, Nederl. Akad. Wetensch. Proc. Ser. A 54 (1951), 50--60, for step (i); Theorem 5.1 of E. R. Canfield, P. Erdős and C. Pomerance, On a problem of Oppenheim concerning "Factorisatio Numerorum" (cited as to appear), for the factorization count in step (v); the prime number theorem.

Bears on

• Problem 830: the problem asks whether there are infinitely many amicable pairs and whether the number of amicable $1\le a\le b\le x$ exceeds $x^{1-o(1)}$. Each such pair is fixed by its smaller member $a$, an amicable number at most $x$, so the Theorem bounds that count above by $x/e^{(\log x)^{1/3}}$ for large $x$ (an observation of this page). This is an upper bound and does not decide either question; the paper says it cannot prove that there are infinitely many amicable numbers, and records Erdős's conjecture $A(x)\gg x^{1-\epsilon}$ for every $\epsilon>0$ against the conjecture of Bratley, Lunnon and McKay that $A(x)=o(\sqrt x)$ (p. 183).