Troupe 2015 number prime factors values sum proper

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theorem_1_3: The Hardy-Ramanujan normal order for the number of distinct prime factors of s(n), the sum of proper divisors: for each eps > 0 the inequality |omega(s(n)) - log log s(n)| < eps log log s(n) holds for all n <= x outside a set of size o(x); the paper's Remark extends it to Omega(s(n)).

theorem_1_4: Second-moment estimate behind the normal order of omega(s(n)): summed over n <= x outside an explicit exceptional set of size o(x), the squares (omega(s(n)) - log log x)^2 total o(x (log log x)^2) as x tends to infinity.

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Troupe, Lee, On the number of prime factors of values of the sum-of-proper-divisors function. J. Number Theory 150 (2015), 120--135, DOI 10.1016/j.jnt.2014.11.014. The copy read for this card is arXiv:1405.3587v3 (14 September 2015, 12 pages), whose labels the card uses. The arXiv record names arXiv's non-exclusive distribution license (arXiv:1405.3587), every other right reserved.

The paper establishes a Hardy-Ramanujan type normal order result for omega(s(n)), where s(n) is the sum of the proper divisors of n. Theorem 1.3 (p. 1) says that, for each fixed eps > 0, the inequality |omega(s(n)) - log log s(n)| < eps log log s(n) fails for only o(x) integers n <= x, and Section 5 shows the same holds with Omega in place of omega; this statement would follow from the Erdos-Granville-Pomerance-Spiro conjecture (Conjecture 1.1, p. 1, that s^{-1}(A) has density zero whenever A does; its image form is EGPS Conjecture 4), but is proved here unconditionally. The result is deduced from Theorem 1.4 (p. 2), the second-moment estimate sum over n <= x outside an exceptional set E(x) of size o(x) of (omega(s(n)) - log log x)^2 = o(x (log log x)^2), using that log log s(n) = log log x + O(1) for all but o(x) values of n <= x. The method writes n = mP with P = P(n) the largest prime factor, so that s(n) = P s(m) + sigma(m). For a prime p not dividing s(m), p | s(n) then puts P in one residue class modulo p (for a pair of primes p, q, modulo pq), and these primes P are counted with a prime number theorem for progressions (Theorem 2.4); this gives the first and second moments of omega(s(n)) over n outside E(x) (Theorem 3.1, Lemma 4.1). Problem 955 is the assertion of Conjecture 1.1 and lists this paper as [Tr15] among its references; the paper does not settle the conjecture.

Source: https://arxiv.org/abs/1405.3587.

Bears on. #955: for each eps > 0, Theorem 1.3 shows that the set of m with |omega(m) - log log m| >= eps log log m, which has density zero by Hardy and Ramanujan's theorem, has a density-zero preimage under s, and its Omega form does the same for the Omega analogue. These are particular density-zero sets; the general assertion, the paper's Conjecture 1.1, is not settled here.

Results to transcribe.

• Theorem 1.3 (p. 1): For each fixed eps > 0, |omega(s(n)) - log log s(n)| < eps log log s(n) fails for only o(x) integers n <= x; a Remark (p. 2), proved in Section 5, gives the same for Omega(s(n)).
• Theorem 1.4 (p. 2): As x -> infinity, sum_{n <= x, n not in E(x)} (omega(s(n)) - log log x)^2 = o(x (log log x)^2), with E(x) the set of Section 2.1 (p. 2), of size o(x) by Lemma 2.2.
• Conjecture 1.1 (p. 1): The Erdős-Granville-Pomerance-Spiro conjecture that s^{-1}(A) has density zero whenever A does; it implies Theorem 1.3, which is proved here unconditionally.
• Lemma 2.1 (p. 2): The sum of omega(s(n)) over n <= x equals, up to o(x log log x), the number of pairs (p, n) with n <= x, p prime, p | s(n) and log log x < p <= x^{1/sqrt(log log x)}.

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