---

Source. Theorem 1.1, p. 2, proved in Section 3 on pp. 3--6, 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. 1--2). $\sigma$ is the sum-of-divisors function and $s(n)=\sigma(n)-n$. Two different positive integers $a,b$ with $s(a)=b$ and $s(b)=a$ form an amicable pair; a positive integer is amicable if it belongs to an amicable pair. $\mathcal{A}$ is the set of amicable numbers and $\mathcal{A}(x)=\mathcal{A}\cap[1,x]$.

Theorem 1.1 (p. 2, quoted). "As $x\to\infty$, we have"

$$\#\mathcal{A}(x)\le x/\exp\Bigl(\bigl(\tfrac12+o(1)\bigr)\sqrt{\log x\log\log\log x}\Bigr).$$

Since $\log\log\log x\to\infty$, this gives $\#\mathcal{A}(x)\le x/e^{\sqrt{\log x}}$ for all sufficiently large $x$, the form stated in the abstract (p. 1). The paper presents the theorem as replacing the exponent $1/3$ in the earlier bound $x/\exp((\log x)^{1/3})$ of Pomerance's 1981 paper (its reference [15]) by $1/2$ (p. 2).

Proof pointer

Section 3, pp. 3--6. With $L=\exp\bigl(\tfrac12\sqrt{\log x\log\log\log x}\bigr)$, the proof discards, in steps (i)--(vii), sets of $n\in\mathcal{A}(x)$ of size at most $x/L^{1+o(1)}$: those with $n$ or $s(n)$ small, with a large $L^2$-smooth divisor or a large squarefull divisor, with a prime above $L$ dividing $\gcd(n,s(n))$, with small cofactors $m,m'$ of the largest primes $p,p'$ of $n=pm$ and $s(n)=p'm'$, with $p$ or $p'$ above $x^{3/4}L$ (the new case, treated through a congruence modulo $\sigma(D)$ for a suitable divisor $D$ of $s(n)$), and with $P(\sigma(m_1))\le L$ for the largest squarefree unitary divisor $m_1$ of $m$ (or likewise $m_1'$ of $m'$), which Lemma 2.1 controls. For the remaining $n$, a prime $r>L$ dividing $\sigma(m_1)$ must also divide $\sigma(\ell^j)$ for a prime power $\ell^j$ dividing $s(n)$, and summing over $r,q,\ell^j,m,p$ gives $O(x(\log x)^5\log\log x/L)$ (p. 6).

Dependencies

Lemma 2.1; de Bruijn's bound for smooth numbers (reference [3]); the elementary inequality (1), p. 4, taken from the proofs of [11, Lemma 3.6] and [12, Lemma 3.3]. Read depth: claims checked; the statement was read clause by clause on p. 2, the proof for its structure on pp. 3--6.

Bears on

• #830: each amicable pair $a<b\le x$ is determined by its smaller member $a\in\mathcal{A}(x)$, so the theorem bounds from above the number of pairs with $a<b$ that the problem's $A(x)$ counts. The problem's condition $a\le b$ also admits $a=b$, that is perfect numbers, which the paper does not count. The bound is of the form $x^{1-o(1)}$, so it is consistent with the conjectured $A(x)>x^{1-o(1)}$; the paper answers neither question of the problem and notes that the infinitude of amicable pairs is unproved (p. 1).