Pratt 2022 irrationality divisor function series erdos kac

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theorem_1: Pratt's unconditional theorem that alpha_4, the sum over n of sigma_4(n)/n! with sigma_4(n) the sum of the fourth powers of the divisors of n, is irrational; the paper gives its value as 42.30104... .

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Kyle Pratt, The irrationality of a divisor function series of Erdős and Kac. Acta Arith. 211 (2023), no. 3, 193–228, DOI 10.4064/aa220927-1-9; arXiv:2209.11124 (2022).

Erdos and Kac conjectured that alpha_k = sum_{n>=1} sigma_k(n)/n! is irrational for every positive k, where sigma_k(n) is the sum of the kth powers of the divisors of n. Irrationality was known for k <= 3 (k = 1, 2 described as not so difficult; k = 3 by Schlage-Puchta and independently Friedlander-Luca-Stoiciu using sieve methods), with general k following from Schinzel's Hypothesis H or a suitable Hardy-Littlewood prime k-tuples conjecture. Theorem 1 of this paper proves unconditionally that alpha_4 = 42.30104... is irrational. The proof is sieve-theoretic, combined with exponential sum estimates, and the author states it pushes those techniques to the limit and that new ideas seem necessary for k >= 5. The paper also places alpha_k in the context of E-functions through the entire functions f_k(z) = sum sigma_k(n) z^n / n!, noting that they do not appear to satisfy any suitable differential equation susceptible to the Siegel-Shidlovskii technique. Theorem 1 is the case k = 4 of Erdos Problem 252, which asks whether sum sigma_k(n)/n! is irrational.

Source: https://arxiv.org/abs/2209.11124. The arXiv record names arXiv's non-exclusive distribution license (arXiv:2209.11124), every other right reserved.

The copy read for this card is arXiv:2209.11124v1 (22 Sep 2022, 28 pp.; 330,708 bytes); its pages 1-3 and the outline of the proof in Section 3 (pp. 4-9) were read for the statements below, and the locators here are its page numbers. The refereed version in Acta Arithmetica (per-article charge at the publisher) was not obtained or compared; the journal record (volume, issue, pages, DOI) was checked against Crossref. For the irrationality of alpha_1 and alpha_2 the introduction cites the Monthly problems of Erdős (Problem 4493) and of Erdős and Kac (Problem 4518), with their solutions by Kelly and by Breusch, and names the Deajim–Siksek criterion for the linear independence of 1, alpha_1, ..., alpha_r under Hypothesis H.

Bears on.

• #252: the problem asks whether sum_n sigma_k(n)/n! is irrational for k >= 1; Theorem 1 proves it irrational for k = 4 and says nothing about any other k.

Results.

• Theorem 1 (p. 1): alpha_4 = sum_{n>=1} sigma_4(n)/n! = 42.30104... is irrational, with no hypothesis.
• E-function remark (p. 2, not a numbered result): the functions f_k(z) = sum sigma_k(n) z^n/n! do not appear to satisfy any suitable differential equation susceptible to the Siegel-Shidlovskii technique.

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