Weingartner 2019 constant factor asymptotic practical numbers

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lemma_2: Weingartner's identity that, for a set B defined by a growth condition theta on successive prime factors, the part of the series of Lemma 1 over the n with q^h exactly dividing n equals (1 - q^{-s}) q^{-sh} times the part over the n with theta(n) at least q, for Re(s) > 1 and, when B(x) = o(x), at s = 1.

lemma_4: Weingartner's explicit bounds for the error terms eta(x) and delta(x) of the Mertens-type sums of log p/(p-1) and log p/p over primes, valid for x >= 2^k with a tabulated constant M_k for each k from 24 to 38.

theorem_1: Weingartner's theorem that the constant c in the asymptotic P(x) ~ cx/log x for the count of practical numbers up to x lies strictly between 1.336073 and 1.336077.

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Andreas Weingartner, The constant factor in the asymptotic for practical numbers. arXiv preprint (2019). arXiv:1906.07819. The copy read for this card is arXiv version 3 (28 Aug 2019). The arXiv record names arXiv's non-exclusive distribution license (arXiv:1906.07819), every other right reserved.

Practical numbers are those n for which every m <= n is a sum of distinct divisors of n, and by the author's earlier work their counting function satisfies P(x) = (cx/log x)(1 + O(log log x/log x)), confirming Margenstern's conjecture P(x) ~ cx/log x. Theorem 1 rigorously encloses the constant, proving 1.336073 < c < 1.336077, a sharp improvement on the author's earlier enclosure 1.311 < c < 1.693; Margenstern's empirical estimate was c approximately 1.341. The constant is the sum of an explicit series over practical numbers n, each term being 1/n times the difference between the sum of log p/(p-1) over primes p <= sigma(n) + 1 and log n, times the product of (1 - 1/p) over the same primes, the whole multiplied by 1/(1 - e^{-gamma}). The improvement comes from a new identity (Lemma 2) used together with the multiplicativity of sigma(n) in place of the earlier extremal-behavior estimate for sigma(n), so the residual gap is almost entirely the error term of Lemma 4 in evaluating the inner prime sum. Lemmas 1 and 2 are stated in the author's earlier general setup, for the set B of integers whose successive prime factors satisfy p_{j+1} <= theta(p_1^{a_1} ... p_j^{a_j}) for an arithmetic function theta, so they apply to other such sets besides the practical numbers (theta(n) = sigma(n) + 1). The paper says nothing about representing a fixed integer t as a sum of distinct divisors of n.

Source: https://arxiv.org/abs/1906.07819.

Bears on. #859: by the definition of a practical number, if N is practical and N >= t, then t is a sum of distinct divisors of every multiple of N; Theorem 1 concerns the constant in the count of the practical numbers themselves and gives nothing about the density d_t of the integers that represent a fixed t.

Results.

• Theorem 1 (p. 1): 1.336073 < c < 1.336077 for the constant in P(x) ~ cx/log x.
• Lemma 2 (p. 2): the new identity, in the general setting, for the part of the sieve series over the n with q^h exactly dividing n.
• Lemma 4 (p. 4): explicit bounds for the error terms of the prime sums of log p/(p-1) and log p/p, for x >= 2^k with 24 <= k <= 38.

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