Weingartner 2015 practical numbers distribution divisors

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corollary_1: For x >= t >= 2 the number D(x,t) of n <= x whose consecutive divisors have ratio at most t equals (x C(t) log t/log xt)(1 + O(1/log x + log^2 t/log^2 x)), where 0 < C_0 <= C(t) = C eta(t) = C + O(1/log t).

theorem_1: For x >= 3 the number P(x) of practical numbers up to x equals (cx/log x)(1 + O(log log x/log x)) for a positive constant c, which proves Margenstern's conjecture.

theorem_2: If theta(1) >= 2 and n <= theta(n) <= An(log 2n)^a(log log 3n)^b with A >= 1, and either 0 <= a < 1, or a = 1 and b < -1, the count B(x) of the integers built under the theta-condition equals (c_theta x/log x)(1 + error) with an explicit error term.

theorem_3: For x >= 1 and t >= 2 the number D(x,t) of n <= x whose consecutive divisors have ratio at most t equals x eta(t) d(v)(1 + O(1/log 2x)), with v = log x/log t and 0 < eta_0 <= eta(t) = 1 + O(1/log t).

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Andreas Weingartner, Practical numbers and the distribution of divisors. The Quarterly Journal of Mathematics 66 (2015), no. 2, 743-758. DOI 10.1093/qmath/hav006. arXiv:1405.2585. The copy read for this card is arXiv:1405.2585v3 (3 March 2015). The arXiv record names arXiv's non-exclusive distribution license (arXiv:1405.2585), every other right reserved.

Theorem 1 proves that the number P(x) of practical numbers up to x equals (cx/log x)(1 + O(log log x/log x)) for a positive constant c, settling Margenstern's conjecture and sharpening Saias's earlier two-sided bound c_1 x/log x <= P(x) <= c_2 x/log x. It is deduced from the far more general Theorem 2, which gives an asymptotic B(x) = (c_theta x/log x)(1 + error) for the count of integers whose prime factorization satisfies p_j <= theta(product of earlier prime powers), whenever theta(1) >= 2 and n <= theta(n) <= A n (log 2n)^a (log log 3n)^b for constants A >= 1, 0 <= a < 1 (error O((log x)^{a-1} (log log x)^b)) or a = 1, b < -1 (error O((log log x)^{b+1})); practical numbers are the case theta(n) = sigma(n) + 1, and Thompson's weakly phi-practical numbers the case theta(n) = n + 2. Writing D(x,t) for the number of n <= x in which every ratio of consecutive divisors is at most t, and v = log x/log t, Theorem 3 gives D(x,t) = x eta(t) d(v) (1 + O(1/log 2x)) for all x >= 1 and t >= 2. It improves the error term of the author's earlier formula D(x,t) = x d(v)(1 + O(1/log t)), proved for x >= t >= exp((log log x)^{5/3+eps}), and removes that lower bound on t; Saias's two-sided estimate D(x,t) asymp x log t/log xt already held for all t >= 2. Corollary 1 turns Theorem 3 into D(x,t) = (x C(t) log t/log xt)(1 + O(1/log x + log^2 t/log^2 x)) for x >= t >= 2. The method is a functional equation (Lemma 3) that writes every m <= x uniquely as nr with n in the counted set and every prime factor of r above theta(n), estimated with Buchstab's function through Tenenbaum's sieve bounds and Saias's two-sided bound for D(x,t), and solved by Laplace transforms against the equation for d(v). For problem 859 this is lower-route context read in full: practical numbers do supply the required subset sums, but these counting theorems concern integers varying over scales, not a fixed target, so they yield no asymptotic for the fixed-target subset-sum density d_t.

Source: https://arxiv.org/abs/1405.2585.

Bears on. #859: if $N$ is practical and $N\ge t$, then $t$ is a sum of distinct divisors of every multiple of $N$; Theorem 1 counts the practical numbers themselves and gives nothing about the density $d_t$ of the integers that represent a fixed $t$. #673: Theorem 1 gives that the practical numbers have density zero, and Corollary 1 at $t=2$ that the integers whose consecutive divisors all have ratio at most $2$, on which $G(n)\ge(\tau(n)-1)/2$, have density zero; neither says anything about $G(n)$ for almost all $n$ or about its average.

Results. Theorem 1 (p. 1); Theorem 2 (p. 2), with its general form Theorem 4 (pp. 11--12) on the same page; Theorem 3 (p. 4), with Corollaries 2 and 3 (p. 4) on the same page; Corollary 1 (p. 4). Corollary 4 (p. 5), on the integers whose largest consecutive-divisor ratio equals $t$, is summarized on the Corollary 1 page; Lemmas 1--15 are proof steps, and the functional equation of Lemma 3 (p. 6) is described on the Theorem 2 and Theorem 3 pages.

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