Pollack 2014 arithmetic properties sum proper divisors sum

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corollary_1_5: As x tends to infinity, the integers n <= x with s(n) < n but s(s(n)) >= s(n) number at most x/exp((1/10 + o(1)) sqrt(log_3 x log_4 x)), a quantitative form of the consequence of the Erdős--Granville--Pomerance--Spiro theorem that s(s(n)) < s(n) for almost all n with s(n) < n.

lemma_2_8: For x >= 3, a natural number q <= x^{1/(2 log_3 x)} and eps > 0, the n <= x outside the exceptional set E(x) with q dividing s(n) number O_eps(x/q^{1-eps}); the paper's key new ingredient for Theorem 1.4.

theorem_1_10: For all x >= 2, pi_beta(x) - pi(x) << x/log x, where pi_beta(x) counts the n <= x whose sum of distinct prime divisors beta(n) is prime; an upper bound of the order that Conjecture 1.9 predicts.

theorem_1_11: For all x >= 2, the number of n <= x for which the sum of proper divisors s(n) is prime is O(x/log x); in particular the preimage of the primes under s has density zero.

theorem_1_4: A quantitative form of the Erdős--Granville--Pomerance--Spiro theorem, the case K = 2 of Erdős's Conjecture 1.3: for x >= 1, the inequality s(s(n))/s(n) - s(n)/n <= (log_2 x)^{-1/4} fails for at most O(x(log_3 x)^2/(log_2 x)^{1/4}) positive integers n <= x.

theorem_1_7: For every real u, the proportion of the integers 1 < n <= x with s(beta(n))/beta(n) <= u tends to D(u), Davenport's distribution function of s(n)/n, where beta(n) is the sum of the distinct prime divisors of n.

theorem_1_8: The natural numbers n whose sum of distinct prime divisors beta(n) is squarefree have asymptotic density 6/pi^2, the density of the squarefree numbers themselves.

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Pollack, Paul, Some arithmetic properties of the sum of proper divisors and the sum of prime divisors. Illinois J. Math. 58 (2014), no. 1, 125--147, doi:10.1215/ijm/1427897171. The copy read for this card is the publisher's PDF, which prints "©2015 University of Illinois" (Illinois Journal of Mathematics, Volume 58, Number 1, Spring 2014), every other right reserved.

Pollack gives a new quantitative proof of the K=2 case of an Erdos conjecture on iterates of the sum-of-proper-divisors function s(n): Theorem 1.4 shows that at most O(x(log_3 x)^2/(log_2 x)^{1/4}) integers n <= x have s(s(n))/s(n) - s(n)/n > (log_2 x)^{-1/4}, a sharper and simpler replacement for the Erdos-Granville-Pomerance-Spiro argument; here log_k is the k-th iterate of log_1 x = max{1, log x}. Corollary 1.5 deduces that at most x/exp((1/10 + o(1)) sqrt(log_3 x log_4 x)) deficient integers n <= x (those with s(n) < n) have s(s(n)) >= s(n). The key new ingredient (Lemma 2.8) is an upper bound, uniform in a wide range of q, for the count of n <= x outside a density-zero exceptional set whose s(n) is divisible by q; the proof of Theorem 1.4 avoids facts about primitive alpha-abundant numbers and for the most part uses only elementary analytic number theory. The same techniques applied to beta(n), the sum of the distinct prime divisors, give Theorem 1.7, that s(beta(n))/beta(n) obeys the Davenport distribution function D(u), and Theorem 1.8, that beta(n) is squarefree on a set of asymptotic density 6/pi^2, the same density as the squarefree numbers. Finally, Theorem 1.10 bounds the number of composite n <= x with beta(n) prime by O(x/log x), and Theorem 1.11 shows that for all x >= 2 the number of n <= x with s(n) prime is O(x/log x). This bears on Erdos problem 955, which asserts that every set of density zero has a density-zero preimage under s: Theorem 1.11 (p. 129) gives that assertion for the one set of the primes.

Source: https://www.pollack-math.net/research.html.

Bears on. #955: Theorem 1.11 shows that the n <= x with s(n) prime number O(x/log x), so the preimage of the primes under s has density zero; this is the problem's assertion for that one set only, and the paper says nothing about other sets of density zero.

Results. Result pages: theorem_1_4, corollary_1_5, theorem_1_7, theorem_1_8, theorem_1_10, theorem_1_11 and lemma_2_8.

• Theorem 1.4 (p. 127): for x >= 1, at most O(x(log_3 x)^2/(log_2 x)^{1/4}) positive integers n <= x have s(s(n))/s(n) - s(n)/n > (log_2 x)^{-1/4}.
• Corollary 1.5 (p. 127): as x -> infinity, at most x/exp((1/10 + o(1)) sqrt(log_3 x log_4 x)) deficient integers n <= x have s(s(n)) >= s(n).
• Theorem 1.7 (p. 128): for every real u, (1/x)#{1 < n <= x : s(beta(n))/beta(n) <= u} tends to the Davenport distribution function D(u) as x -> infinity.
• Theorem 1.8 (p. 128): beta(n) is squarefree for a set of n of asymptotic density 6/pi^2, the density of the squarefree integers themselves.
• Theorem 1.10 (p. 129): for all x >= 2, pi_beta(x) - pi(x) << x/log x, where pi_beta(x) counts the n <= x with beta(n) prime.
• Theorem 1.11 (p. 129): for all x >= 2, the number of n <= x for which s(n) is prime is O(x/log x).
• Lemma 2.8 (p. 133): with x >= 3 as throughout Section 2.2, let q be a natural number with q <= x^{1/(2 log_3 x)}, and let e > 0. Among the n <= x outside E(x) = {n <= x : P(n) <= x^{1/log_3 x} or P(n)^2 | n}, with P(n) the largest prime factor of n, at most O_e(x/q^{1-e}) have q dividing s(n); by Lemma 2.6, #E(x) << x/(log_2 x)^4.

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