Weingartner 2025 schinzel szekeres function

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Weingartner, Andreas, The Schinzel-Szekeres function. Res. Number Theory 11 (2025), no. 3, Paper No. 63, 32 pp. DOI 10.1007/s40993-025-00643-9 (Crossref record read).

Weingartner derives sharp asymptotics for distribution functions attached to the Schinzel-Szekeres function F(n) = max{d P^-(d) : d | n, d > 1}. Theorem 1 shows the counting function A(x) = #{n : F(n) <= x} satisfies A(x) = a x / log x (1 + O(1/log x)) with the explicit constant a = 1.53796..., improving Saias's A(x) asymptotic order to a genuine asymptotic. Three applications follow. Corollary 1 gives f(n) > 0.76898 n / log n for the longest simple path in the divisor graph of order n, improving the previous 0.37 n / log n. For Erdos's problem on how large sum_{n in S} 1/n can be for S contained in {2,...,x} with pairwise lcm exceeding x, Theorem 2 estimates the reciprocal sum over the Schinzel-Szekeres set B(x), giving sum 1/n = 1 - delta/log x + O((log x)^{-3/2}) with delta = 0.560..., and Theorem 3 improves the resulting lower bound to R(x) >= 1 - kappa/log x + O((log x)^{-3/2}) with kappa = 0.543..., by modifying B(x) using the two known cases R(5) = 31/30 and R(11) = 4699/4620. Question 1 asks whether R(x) = 1 - (kappa + o(1))/log x. This is the current best lower bound on R(x), the maximal reciprocal sum under the lcm condition, which is the first question of problem 542, not problem 784. The third application is the small sieve of problem 784: with H(x) the least number of n up to x divisible by no member of a set of integers greater than 1 with reciprocal sum at most 1, Theorem 4 gives H(x) <= a e^{-delta} x / log x + O(x/(log x)^{3/2}) with a e^{-delta} = 0.878..., Corollary 2 gives H(x) < 0.879 x / log x for large x, and Question 2 asks whether this is the asymptotic; Theorem 5 extends this to the budget z, with H(x, 1 + mu/log x) <= a e^{-delta-mu} x / log x + O(x/(log x)^{3/2}) for constant mu > -delta, and, for each fixed Z > 1, the exact order x^{e^{1-z}} / log x for H(x,z) (bounded above and below by constant multiples of it) uniformly for 1 <= z <= Z, x >= 2, improving Ruzsa's logarithmic asymptotic log H(x,z)/log x -> e^{1-z}.

Source: https://arxiv.org/abs/2310.13038. The held folder-name PDF is arXiv:2310.13038v2 (stamped 13 June 2025, 29 pages), fetched from arXiv; the labels above are those of this version. The arXiv record (https://arxiv.org/abs/2310.13038, read 2026-10-02) names the Creative Commons Attribution 4.0 license.

Bears on. #784: Theorem 5 gives the exact order x^{e^{1-z}} / log x of the least unsifted count for reciprocal budget z, uniformly for 1 <= z <= Z with Z > 1 fixed, which answers the question yes at C = 1 and no for fixed C > 1; at z = 1, Theorem 4 and Corollary 2 give only the upper bound H(x) < 0.879 x / log x for large x, and Question 2 asks for the asymptotic.

Results to transcribe.

• Theorem 1: A(x) = #{n : F(n) <= x} equals a x / log x (1 + O(1/log x)) with a = 1.53796... given by an explicit integral.
• Corollary 1: The longest simple path in the divisor graph of order n has f(n)

  0.76898 n / log n for all large n.

• Theorem 2: For the Schinzel-Szekeres set B(x), sum_{n in B(x)} 1/n = 1 - delta/log x + O((log x)^{-3/2}) with 0.560374 < delta < 0.560579.
• Theorem 3: R(x) >= 1 - kappa/log x + O((log x)^{-3/2}) with 0.543595 < kappa < 0.543804, using a modified set B'(x).
• Question 1: Asks whether R(x) = 1 - (kappa + o(1))/log x as x tends to infinity.
• Theorem 4: H(x) <= H^*(x) = a e^{-delta} x / log x + O(x/(log x)^{3/2}) for x >= 2, with 0.877992 < a e^{-delta} < 0.878171.
• Corollary 2: H(x) < 0.879 x / log x for all sufficiently large x.
• Question 2: Asks whether H(x) is asymptotic to a e^{-delta} x / log x.
• Theorem 5: For constant mu > -delta and x >= 2, H(x, 1 + mu/log x) <= H^*(x, 1 + mu/log x) = a e^{-delta-mu} x / log x + O(x/(log x)^{3/2}); for fixed Z > 1, uniformly for 1 <= z <= Z and x >= 2, H(x,z) has exact order x^{e^{1-z}} / log x, improving Ruzsa's log H(x,z)/log x -> e^{1-z}.