Lichtman 2020 almost primes banks martin conjecture

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theorem_2_1: Lichtman's theorem that the Erdős sum over the integers with exactly six prime factors, counted with repetition, is smaller than the sum for every other number of prime factors, so the Banks–Martin monotonicity conjecture fails.

theorem_2_2: Lichtman's theorem that the Erdős sum over the integers with exactly k prime factors tends to 1 as k grows, and the squarefree analogue tends to 6/π², with the error O(k^{-1/2+ε}) for every ε > 0 proved in Theorem 4.1.

theorem_5_3: Lichtman's exponentially decaying upper bound for the Erdős sum over the integers with exactly k prime factors: the sum is at most 1 + O(k 2^{-k/2}), proved by the prime zeta function method.

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Lichtman, Jared Duker, Almost primes and the {B}anks-{M}artin conjecture. J. Number Theory 211 (2020), 513--529, doi:10.1016/j.jnt.2019.11.006. The copy read for this card is arXiv:1909.00804v2 (18 December 2019). The arXiv record names arXiv's non-exclusive distribution license (arXiv:1909.00804), every other right reserved.

For N_k the set of integers with exactly k prime factors counted with repetition, the paper studies f(N_k) = sum over n in N_k of 1/(n log n); each N_k is a primitive set, and Erdos proved f(A) bounded uniformly over all primitive sets A (p. 1). Extending Cohen's prime-zeta-function method to the k-almost prime zeta functions P_k(s) (Proposition 3.1, p. 4), the author computes f(N_k) and its squarefree analog f(N*k) to 20 digits for k = 2 to 10 (Figure 2, p. 2) and observes that f(N_k) decreases for k <= 6 and increases thereafter (p. 3), contrary to the Banks-Martin conjecture that f(N_k) > f(N{k+1}) for all k >= 1. Theorem 2.1 (p. 3, proved as Theorem 5.5, p. 11) proves f(N_6) < f(N_k) for every positive integer k other than 6; its proof takes k <= 20 from the computed values of Figures 2 and 3. Theorem 2.2 (p. 3) proves f(N_k) tends to 1 and f(N*_k) to 6/pi^2 as k grows, in the quantitative form f(N_k) = 1 + O(k^{-1/2+eps}) and f(N*_k) = 6/pi^2 + O(k^{-1/2+eps}) for any eps > 0 (Theorem 4.1, p. 6), by partial summation and the Sathe-Selberg theorem in the critical range x about e^{e^k}. Theorem 5.3 (p. 10) gives the one-sided bound f(N_k) <= 1 + O(k 2^{-k/2}) by the zeta method. The paper presents the Banks-Martin conjecture as an extension of Erdos's conjecture f(A) <= f(N_1) = 1.636... for primitive sets A (p. 1), and does not mention Erdos problem 1196.

Source: https://arxiv.org/abs/1909.00804. Page numbers on the result pages are those of the arXiv edition named above.

Read status: claims checked for Theorems 2.1, 2.2, 4.1 and 5.3, read clause by clause on the page images of the arXiv edition; the proofs of Theorems 4.1, 5.3 and 5.5 followed for their structure. The numerical values of Figures 2 and 3, on which the case k <= 20 of Theorem 2.1 rests, were not reproduced. Nothing here is independently reviewed.

Bears on. #1196: the paper does not state the problem. Its Theorem 2.2 gives primitive sets N_k, with least element 2^k, whose sums f(N_k) tend to 1, the lower-bound example the problem's page reports from the site's commentary; the paper proves nothing about the upper bound the problem asks for.

Results.

• Theorem 2.1 (p. 3; Theorem 5.5, p. 11): f(N_6) < f(N_k) for every positive integer k other than 6.
• Theorem 2.2 (p. 3) and Theorem 4.1 (p. 6): f(N_k) tends to 1 and f(N*_k) to 6/pi^2, each with error O(k^{-1/2+eps}) for any eps > 0.
• Theorem 5.3 (p. 10): f(N_k) <= 1 + O(k 2^{-k/2}).

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