Pomerance 2015 amicable numbers

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lemma_2_1: Pomerance's lemma that, for each fixed eps > 0, the number of squarefree n <= x whose sigma(n) has largest prime factor at most y is at most x exp(-(1+o(1)) u log log u) as u = log x/log y tends to infinity, for (log log x)^{1+eps} <= y <= x.

theorem_1_1: Pomerance's theorem that, as x tends to infinity, the number of amicable numbers up to x is at most x/exp((1/2+o(1)) (log x log log log x)^{1/2}), so at most x/exp((log x)^{1/2}) for all large x.

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Pomerance, Carl, 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. The copy read for this card is the author's manuscript from the author's page (https://math.dartmouth.edu/~carlp/), which states no terms for the papers it links, and the file prints no notice; the term is unstated.

Pomerance sharpens the known upper bound for the counting function of the amicable numbers: Theorem 1.1 states that as x tends to infinity, #A(x) <= x/exp(((1/2)+o(1)) sqrt(log x log log log x)), which in particular gives #A(x) <= x/e^{sqrt(log x)} for all sufficiently large x, replacing the exponent 1/3 of Pomerance's 1981 paper by 1/2. The argument follows the plan of the 1981 proof with new elements: a separate treatment when an amicable n is divisible by a very large prime p, reminiscent of work on Lehmer's problem for Euler's function, while for smaller largest prime factor p one shows that usually a fairly large prime divides sigma(n/p). Lemma 2.1 supplies the needed estimate for the number of squarefree n <= x with P(sigma(n)) <= y, stated as following by small cosmetic changes from the bound x exp(-(1+o(1)) u log log u) of Banks, Friedlander, Pomerance and Shparlinski for the number of n <= x with P(phi(n)) <= y. The paper reviews, in simplified form, the earlier upper bounds from Rieger's x/(log log log log x)^{1/2} to x/exp((log x)^{1/3}), and notes that about 12 million amicable pairs are known but their infinitude is unproved.

Source: https://math.dartmouth.edu/~carlp/.

Bears on. #830: Theorem 1.1 bounds the number of amicable numbers up to $x$ from above, and so the number of pairs $a<b\le x$ that the problem's $A(x)$ counts, each pair being determined by its smaller member; 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 $x^{1-o(1)}$ and is consistent with the conjectured lower bound; the paper answers neither question of the problem.

Read status. Claims checked: Theorem 1.1 (p. 2) and Lemma 2.1 (p. 3) were read clause by clause; the proof of Theorem 1.1 (pp. 3--6) was read for its structure and not checked step by step. The paper writes out no proof of Lemma 2.1, deferring to [2, Theorem 3.1]. Labels and pages are those of the manuscript read (pp. 1--7); its page-to-print mapping onto the published pp. 321--327 was not checked.

Results. Theorem 1.1 (p. 2; proof pp. 3--6); Lemma 2.1 (p. 3; no proof written out).

No file of this source is held: no license on record permits its redistribution, and the card cites the edition it names above.