Gorodetsky 2024 erdos sums almost primes

../

proposition_4_1: Gorodetsky, Lichtman and Wong's probabilistic estimate for the Erdős sum of k-almost primes, f_k = 1 + O(k/2^(k/4)), weaker than their Theorem 1.2 but proved by comparing f_k with e^gamma times an iterated integral.

theorem_1_1: Gorodetsky, Lichtman and Wong's monotonicity theorem: for y >= 2 and k sufficiently large, the Erdős sums of k-almost primes increase in k, contrary to the Banks-Martin conjecture, while the sums restricted to integers without prime factors at most y decrease, as Banks and Martin conjectured.

theorem_1_2: Gorodetsky, Lichtman and Wong's asymptotic for the Erdős sum f_k of the integers with exactly k prime factors counted with multiplicity: for all k >= 1, f_k = 1 - 2^(-k)(a_1 k^2 + O(k log(k+1))), where a_1 = (d log 2)/4 = 0.0656... with d = 0.37869... the constant of the paper's (1.1).

theorem_1_3: Gorodetsky, Lichtman and Wong's asymptotic for the Erdős sum f_{k,y} of the k-almost primes with no prime factor at most y: for y >= 2, uniformly for k >= 1, f_{k,y} equals the product of (1 - 1/p) over p <= y, plus c_y d_y / 2^k, plus O_y(k^3/3^k).

theorem_1_6: Gorodetsky, Lichtman and Wong's estimate for the iterated integrals I_k of 1/(1 + x_1(1 + x_2(... (1 + x_k) ...))) over the unit cube: the integrals satisfy I_k = e^(-gamma) + O(2^(-k)), the special case c_j = 1 of their Theorem 4.8.

---

Gorodetsky, Ofir and Lichtman, Jared Duker and Wong, Mo Dick, On {E}rdős sums of almost primes. C. R. Math. Acad. Sci. Paris 362 (2024), 1571--1596, doi:10.5802/crmath.650. The copy read for this card is arXiv:2303.08277v2 (12 May 2024). The arXiv record names arXiv's non-exclusive distribution license (arXiv:2303.08277), every other right reserved.

The paper studies the Erdos sums f_k = sum over n with Omega(n) = k of 1/(n log n), bounded by Erdos in 1935 and maximized at k = 1 by Zhang. Theorem 1.1 shows that for k large the sums f_k are increasing, contradicting the 2013 Banks-Martin conjecture that they decrease, while the y-truncated sums f_{k,y} (restricted to integers with all prime factors above y) do decrease for any y >= 2, confirming that half of the conjecture in the large-k range. Theorem 1.2 gives, for all k >= 1, the asymptotic f_k = 1 - 2^{-k}(a_1 k^2 + O(k log(k+1))) with a_1 = (d log 2)/4 = 0.0656... and d = 0.37869... the constant of (1.1), an exponential refinement of the Sathe-Selberg bound f_k = 1 + O_eps(k^{eps - 1/2}). The method combines real and complex analysis on the generating Dirichlet series. A second, probabilistic argument tied to the Dickman distribution gives only the weaker f_k = 1 + O(k/2^{k/4}) (Proposition 4.1): it compares f_k with e^gamma I_{[k/4]} for iterated integrals I_k over [0,1]^k, and Theorem 1.6 proves I_k = e^{-gamma} + O(2^{-k}). Erdos problem 1196 asks whether every primitive set A in [x, infinity) has sum over A of 1/(a log a) below 1 + o(1); the k-almost primes form a primitive set with least element 2^k, and Theorem 1.2 shows their sums f_k tend to 1 from below, so the constant 1 there cannot be lowered.

Source: https://arxiv.org/abs/2303.08277.

Bears on.

• #1196: the k-almost primes form a primitive set lying in [2^k, infinity), and Theorem 1.2 gives its sum of 1/(n log n) as 1 - (a_1 + o(1)) k^2/2^k: below 1 for large k and tending to 1, so these sets approach the problem's constant 1 from below (Proposition 4.1 gives the weaker 1 + O(k/2^{k/4})). The paper does not pose or answer the problem.

Results.

• Theorem 1.1 (p. 2): For y >= 2 and k sufficiently large, f_{k-1} < f_k and f_{k-1,y} > f_{k,y}: the Erdos sums increase, contrary to the Banks-Martin conjecture, and the sifted ones decrease.
• Theorem 1.2 (p. 2): For all k >= 1, f_k = 1 - 2^{-k}(a_1 k^2 + O(k log(k+1))) with a_1 = (d log 2)/4 = 0.0656... and d = 0.37869... as in (1.1).
• Theorem 1.3 (p. 2): For y >= 2, uniformly for k >= 1, f_{k,y} = prod_{p <= y}(1 - 1/p) + c_y d_y/2^k + O_y(k^3/3^k), with c_y and d_y given by (1.2) and (1.3).
• Theorem 1.6 (p. 3): The iterated integrals I_k of (1.5) satisfy I_k = e^{-gamma} + O(2^{-k}); the general form with an innermost weight c_j is Theorem 4.8 (p. 22).
• Proposition 4.1 (p. 17): The probabilistic argument gives f_k = 1 + O(k/2^{k/4}).

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