Schlagepuchta 2006 irrationality number theoretical series

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lemma_p2: Schlage-Puchta's sieve lemma, taken from Halberstam and Richert's Theorem 7.4: the primes p <= x whose shifts (p+1)/2 and (p+2)/3 have least prime factor above x^{1/9} number at least of order x/log^3 x; it replaces Hypothesis H in the unconditional proof for k = 3.

theorem_p1: Schlage-Puchta's two-part theorem on S_k, the sum over n >= 1 of sigma_k(n)/n!: under Schinzel's Hypothesis H every S_k is irrational, and S_3 is irrational with no hypothesis.

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Schlage-Puchta, J.-C., The irrationality of a number theoretical series. Ramanujan J. 12 (2006), no. 3, 455–460, DOI 10.1007/s11139-006-0154-3; arXiv:1105.1452 (v1, 7 May 2011).

Let sigma_k(n) be the sum of the k-th powers of the divisors of n and set S_k = sum_{n >= 1} sigma_k(n)/n!. Erdos and Straus had shown S_0 and S_1 irrational and Erdos and Kac had shown S_2 irrational, and Erdos asked whether S_k is irrational for all k, which is Problem 252. The paper's single Theorem has two parts: (1) if Schinzel's conjecture H holds then S_k is irrational for every k in N, and (2) S_3 is irrational unconditionally. The conditional proof assumes S_k = a/b rational, notes that (n-1)! S_k then forces sum_{nu >= n} sigma_k(nu)/(nu)_{nu-n+1} to be an integer, and uses primes q = 1 mod k!^k for which (q+i)/(i+1) is prime for all i <= k, supplied by conjecture H, to derive two incompatible estimates on the distance to the nearest integer, ending in the contradiction ||q_1^{k-1}/p^k|| < q_1^{-1+eps} for arbitrarily large q_1 with p fixed. For k = 3 conjecture H is replaced by a sieve lemma, which the paper derives from Halberstam and Richert Theorem 7.4, that the number of primes p <= x with least prime factors of (p+1)/2 and (p+2)/3 both exceeding x^{1/9} is at least of order x/log^3 x. For such primes q the near-integer estimate of part (1) for the first three terms of the tail and the fractional parts {sigma_3(q)/q} = 1/q and {sigma_3(q+1)/(q(q+1)) - sigma_3(q+1)/(q+1)^2} = 7/8 + O(q^{-1/3}) give, with n = (q+1)/2, at least of order x/log^3 x integers n <= x with ||9 sigma_3(n)/(4n^2) + 19/216|| << n^{-1/3} and the sieve conditions. An upper count of such n, split by the number of prime factors of n and using the Erdos-Turan inequality with van der Corput estimates, gives O(x log log x/log^4 x), a contradiction that proves S_3 irrational unconditionally. The paper prints 7/8 and 19/216; direct computation gives 1/8 and 35/216, and the count works with either constant.

Not the same paper as the author's The irrationality of some number theoretical series, Acta Arith. 126 (2007), 295-303 (arXiv:1105.1451), filed as schlagepuchta_2011_irrationality_number_theoretical_series, which proves the rational linear independence of 1 and the sums of p_n^k/n! and does not concern sigma_k.

Source: https://arxiv.org/abs/1105.1452. The arXiv record names arXiv's non-exclusive distribution license (arXiv:1105.1452), every other right reserved.

The copy read for this card is the arXiv posting of 7 May 2011 (arXiv:1105.1452v1, 5 pp.; 111,970 bytes) of the 2006 paper; its first page was read for the theorem and the attribution sentence, and all five pages were later checked on the page images for the proof summary above. The journal version was not compared; the journal record (volume, issue, pages, DOI) was checked against Crossref. The same author's later paper with a nearly identical title, The irrationality of some number theoretical series, Acta Arith. 126 (2007), no. 4 (arXiv:1105.1451), is a different paper, filed as schlagepuchta_2011_irrationality_number_theoretical_series (its slug year is the arXiv posting date); it concerns factorial series with prime-power numerators and bears on Problem 251, not on this one.

Bears on. #252: the problem asks, for each k >= 1, whether the sum of sigma_k(n)/n! is irrational. Theorem (p. 1) part (2) answers yes for k = 3, unconditionally, and part (1) answers yes for every k only under Schinzel's Hypothesis H, which is unproved; the Lemma (p. 2) is the sieve input to part (2) and says nothing about the series by itself.

Results. Page numbers are those of the arXiv posting read (pp. 1--5).

• Theorem (p. 1, unnumbered; proof of part (1) pp. 1--2, of part (2) pp. 2--5): (1) if Schinzel's conjecture H is true, S_k = sum_{n >= 1} sigma_k(n)/n! is irrational for every k in N; (2) S_3 is irrational, unconditionally. The page also gives Hypothesis H as the paper states it (p. 1).
• Lemma (p. 2, unnumbered): the number of primes p <= x for which the least prime factors of (p+1)/2 and (p+2)/3 both exceed x^{1/9} is at least of order x/log^3 x, following from Halberstam-Richert Theorem 7.4.

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