Erdos 1974 irrationality certain series

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corollary_2_10: States that positive integers b_n over nondecreasing a_n, with increments b_(n+1) minus b_n of size at most o(a_n) and a_n over b_n tending to zero along a subsequence, give an irrational series, which reproves the irrationality of the sum of p_n over n factorial from the prime gap bound.

theorem_2_1: States that a series of integers b_n over the products a_1 through a_n, with b_n small against a_(n-1) a_n, is rational exactly when B b_n equals c_n a_n minus c_(n+1) for integers c_n with |c_(n+1)| below a_n over two.

theorem_3_1: States that for a monotonic sequence of positive integers a_n with p_n of size o(a_n squared) and a_n over p_n tending to zero along a subsequence, the sum of p_n over the products a_1 through a_n is irrational; with a_n equal to n it reproves the irrationality of the sum of p_n over n factorial.

theorem_3_7: States that for monotone positive integers a_n above n to the one half plus delta, the numbers one, the sum of phi(n) over a_1 through a_n, the sum of sigma(n) over a_1 through a_n and the sum over a_1 through a_n of any small integer numerators, infinitely many of them nonzero, are rationally independent; with a_n equal to n this gives the case k equal to one of problem 252.

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P. Erdős and E. G. Straus, On the irrationality of certain series, Pacific J. Math. 55 (1974), no. 1, 85--92 (received 16 April 1974); Zbl 0279.10026. Open access at Project Euclid, record https://projecteuclid.org/euclid.pjm/1102911140.

The copy read for this card is the Project Euclid scan: eight physical pages, printed pp. 85--92, head "PACIFIC JOURNAL OF MATHEMATICS Vol. 55, No. 1, 1974". Provenance: fetched from https://projecteuclid.org/journalArticle/Download?urlId=pjm%2F1102911140, 571,724 bytes. The scan's OCR text layer garbles the formulas; the statements below were read on the page images of printed pp. 85--89, and the proof of Theorem 3.7 on pp. 89--91 was read in the text layer only. The scan prints no notice on pp. 85--86 or 91--92; the journal's issue page, which lists the article, shows "© Copyright 1974 Pacific Journal of Mathematics. All rights reserved." (https://msp.org/pjm/1974/55-1/index.xhtml), every other right reserved.

Three papers share this title. This paper, the 1957 Indag. Math. note erdos_1957_irrationality_certain_series and the Math. Student 36 (1968) note filed as erdos_1969_irrationality_certain_series are all called "On the irrationality of certain series". This is the "[Er-Str (74)]" of the 1980 Erdős–Graham monograph.

Contents

Throughout, $\{a_n\}$ and $\{b_n\}$ are integer sequences and the series is $\sum_{n\ge1}b_n/(a_1\cdots a_n)$ (the paper's (2.3)).

• Theorem 1.1 (p. 85) restates the result of the authors' earlier paper [2], Pacific J. Math. 36 (1971), 635--646, filed as erdos_1971_number_theoretic_results: both $\sum\varphi(n)/(a_1\cdots a_n)$ and $\sum\sigma(n)/(a_1\cdots a_n)$ (the paper's (1.2)) are irrational whenever the positive integers $a_n$ are monotonic and satisfy $a_n\ge n^{11/12}$ from some point on. The authors conjecture that monotonicity alone suffices and note that $a_n=\varphi(n)+1$ or $a_n=\sigma(n)+1$ show some condition is needed.
• Theorem 2.1 (pp. 85--86): for integers $b_n$ and positive integers $a_n$ with $a_n>1$ for large $n$ and $|b_n|/(a_{n-1}a_n)\to0$, the series is rational if and only if there are a positive integer $B$ and integers $c_n$ with $Bb_n=c_na_n-c_{n+1}$ and $|c_{n+1}|<a_n/2$ for all large $n$; with the Remark on p. 87.
• Corollary 2.10 (p. 87): under the hypotheses of Theorem 2.1 with $b_n>0$, $a_{n+1}\ge a_n$, $\lim(b_{n+1}-b_n)/a_n\le0$ and $\liminf a_n/b_n=0$, the series is irrational.
• Theorem 3.1 (pp. 87--88): for a monotonic sequence of positive integers $a_n$ with $\lim p_n/a_n^2=0$ and $\liminf a_n/p_n=0$, $\sum p_n/(a_1\cdots a_n)$ is irrational.
• Theorem 3.7 (pp. 88--91): for a monotonic sequence of positive integers with $a_n>n^{1/2+\delta}$, the numbers $1$, $\sum\varphi(n)/(a_1\cdots a_n)$, $\sum\sigma(n)/(a_1\cdots a_n)$ and $\sum d_n/(a_1\cdots a_n)$ with $|d_n|<n^{1/2-\delta}$, $d_n\ne0$ infinitely often, are rationally independent. Its input is Selberg's theorem on primes in almost all short intervals, quoted as Theorem 3.10 (p. 90).

References (p. 92): [1] Erdős, Enseignement Math. 4 (1958), 93--100; [2] Erdős and Straus, Pacific J. Math. 36 (1971), 635--646; [3] A. Selberg, Arch. Math. Naturvid. 47 (1943), 87--105.

Relations

• The prime factorial series. With $a_n=n$ (monotone; $a_n>1$ for $n\ge2$), Theorem 3.1 gives $\sum p_n/n!$ irrational using only $p_n\sim n\log n$; Corollary 2.10 with $a_n=n$, $b_n=p_n$ gives the same from the gap bound $p_{n+1}-p_n=o(n)$. Both reprove the case $k=1$ of Erdős 1958, cited as [1]. Neither extends to $\sum p_n^k/n!$ for $k\ge2$, where $b_n/(a_{n-1}a_n)\to0$ fails.
• Problem 252 for $k=1$. With $a_n=n$, Theorem 3.7 gives the rational independence of $1$, $\sum\varphi(n)/n!$ and $\sum\sigma(n)/n!$, in particular the irrationality of $\sum\sigma(n)/n!$, the case $k=1$ of problem 252; the irrationality alone is already Theorem 1.1, that is the 1971 paper, with $a_n=n\ge n^{11/12}$. Nothing in the paper concerns $\sigma_k$ for $k\ge2$.
• Formalization. The Archive of Formal Proofs lists an Isabelle entry "Irrationality Criteria for Series by Erdős and Straus" by Angeliki Koutsoukou-Argyraki and Wenda Li (12 May 2020, https://isa-afp.org/entries/Irrational_Series_Erdos_Straus.html), whose abstract says it formalizes Theorem 2.1, Corollary 2.10 and Theorem 3.1 and depends on the entry "Elementary Facts About the Distribution of Primes". Only the entry page was read; the formal statements and their hypotheses were not inspected, so whether the formal Theorem 3.1 yields $\sum p_n/n!$ for $a_n=n$ is not recorded here.

Compiled scope

Statements were read on the page images; the proofs of Theorem 2.1, Corollary 2.10 and Theorem 3.1 were read for structure and are summarized on the result pages, and the proof of Theorem 3.7 from the text layer only. No proof is rewritten in full and none has been independently reviewed.

Bears on. #251 (context: Theorem 3.1 and Corollary 2.10 reprove the $k=1$ theorem cited on the problem page and Theorem 3.1 treats the monotone relatives of its series), #252 (Theorem 3.7 and the restated Theorem 1.1 with $a_n=n$ give the case $k=1$).

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