Friedlander 2007 irrationality divisor function series

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Friedlander, J. B., Luca, F. and Stoiciu, M., On the irrationality of a divisor function series. Integers 7 (2007), #A31, 9 pp.; received 12/8/06, revised 3/20/07, accepted 6/12/07, published 7/3/07, as printed on the header; DOI 10.5281/zenodo.8288725 (the journal's Zenodo deposit, checked against DataCite).

Erdos and Kac proved that the sum over n >= 1 of sigma_k(n)/n! is irrational for k = 1 and 2; Erdős's 1988 survey (the paper's [5]) states that the method does not seem to extend to k >= 3 (the problem is B14 in Guy). Theorem 1 of this paper proves unconditionally that the series is irrational for k = 3, and Theorem 2 shows the prime k-tuples conjecture (Conjecture 1) implies irrationality for every k >= 4. The method is the classical one of assuming the sum is A/B, multiplying by (n-1)! for a well-chosen large n, and showing the fractional part is trapped strictly between 0 and 1; for k = 3 n is taken to be a prime p with (p+1)/2 almost prime, supplied by a version of Chen's theorem stated as Theorem 3, which gives many primes p = 1 (mod a) in [x/2, x] with (p+1)/2 a prime or a product of two primes exceeding x^{1/10}. For problem 252 this is the paper that settles the k = 3 case of the Erdos-Kac divisor-series irrationality question and reduces the remaining cases to prime k-tuples.

Source: http://math.colgate.edu/~integers/vol7.html; PDF https://math.colgate.edu/~integers/h31/h31.pdf. The file prints no license line; the journal's site states "All works of this journal are licensed under a Creative Commons Attribution 4.0 International License" (https://math.colgate.edu/~integers/, read 2026-10-02): the Creative Commons Attribution 4.0 license, by the journal's undated site-wide statement.

The retained PDF (9 pp.; 190,241 bytes) was read for the statements below. Theorem 2 (p. 3) is stated for every positive integer k under "the Prime k-tuples Conjecture (see [3, 7, 9]), which is due to Dickson", given as Conjecture 1: for k >= 2 and integers a_i > 0 and b_i such that for every prime p some n makes the product of the a_i n + b_i not divisible by p, infinitely many positive n make every a_i n + b_i prime. The proof of Theorem 2 (Section 3) is written for k >= 4, the cases k <= 3 being covered by Theorem 1 and the short proofs for k = 0, 1, 2 in the introduction; the abstract phrases the conditional result as holding "on the prime k-tuples conjecture for k >= 4". A "Note Added: March 2007" records that Schlage-Puchta's paper [10] obtained the same two results independently, with a rather different sieve proof for k = 3 and a similar conditional proof for larger k. The introduction attributes k = 1 and k = 2 to Erdős and Kac [4], cited as Problem 4518, Amer. Math. Monthly 61 (1954), 264.

Bears on. #252: Theorem 1 settles the case k = 3 unconditionally, and Theorem 2 gives every k >= 4 only under the prime k-tuples conjecture.

Results to transcribe.

• Theorem 1: The series sum_{n>=1} sigma_3(n)/n! is irrational, unconditionally.
• Theorem 2: The prime k-tuples conjecture (Conjecture 1, Dickson's form for linear polynomials) implies sum_{n>=1} sigma_k(n)/n! is irrational; stated for every positive integer k, proved in Section 3 for k >= 4.
• Theorem 3 (p. 3; attributed to Chen): For any integer a there is x_a such that for x > x_a the interval [x/2, x] contains >> x a/(phi(a)^2 (log x)^2) primes p = 1 (mod a) for which (p+1)/2 is a prime or a product of two primes each exceeding x^{1/10}; the main tool for Theorem 1.