Erdos 1957 irrationality certain series

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lemma_1: States the criterion that a series of nonnegative integers over t to the k is irrational when the coefficients have bounded mean and an infinite support of vanishing lower density.

lemma_4: States the criterion that a series of signed integer coefficients over t to the k is irrational under polynomial growth, sparse support along a sequence and an interlacing condition, with the sharper Lemma 4′ stated without proof.

remark_p212: Restates as unproved the irrationality of the totient, divisor-sum and distinct-prime-factor series over t to the n and proves nothing about them.

remark_p213: Records the Erdős–Kac conjecture that the sum of sigma_k(n) over n factorial is irrational for every k, proved for k equal to one and two.

theorem_1: Proves that the sums of one over t to the phi(n) and one over t to the sigma(n) are irrational for every integer base t above one; an exponent variant, not the totient or divisor-sum series of problems 249 and 250.

theorem_2: Proves that the sum of one over t to the n_k satisfies no integer polynomial equation of degree at most l when n_k over k to the l has limit superior infinity, and states the algebraicity question of problem 247.

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P. Erdős, On the irrationality of certain series, Nederl. Akad. Wetensch. Proc. Ser. A 60 = Indag. Math. 19 (1957), no. 2, 212--219; DOI 10.1016/s1385-7258(57)50028-0 (Crossref record read); communicated by J. Popken at the meeting of 29 December 1956.

The copy read for this card is the Rényi archive scan (item 1957-07), whose head reads "Reprinted from Proceedings, Series A, 60, No. 2 and Indag. Math., 19, No. 2, 1957"; its eight physical pages are printed pp. 212--219. Provenance: fetched from https://users.renyi.hu/~p_erdos/1957-07.pdf on 2026-09-17 (UTC), 968,398 bytes. The scan carries an OCR text layer that garbles every formula; the statements below were read on the page images. No copyright line is printed on the offprint, whose head reads "Reprinted from Proceedings, Series A, 60, No. 2 and Indag. Math., 19, No. 2, 1957" (pp. 218--219 print none); the hosting archive's site footer speaks for the site, not the paper ("(C) 2005-2007 All rights reserved. All material on this site is for scientifics purposes only.", https://users.renyi.hu/~p_erdos/, read 2026-10-02); the KNAW digital library hosts the Proceedings only for 1895--1950 and states no copyright or license (https://dwc.knaw.nl/toegangen/digital-library-knaw/, read 2026-10-02); the publisher's page could not be read (ScienceDirect answered HTTP 403), and the Crossref record for DOI 10.1016/s1385-7258(57)50028-0, read 2026-10-07, names only the publisher's own terms, Elsevier's text-and-data-mining user license and, from 2015-02-13, its open-archive user license (elsevier.com/open-access/userlicense/1.0/), and no Creative Commons license, every other right reserved.

Three papers share this title. This 1957 note, the Math. Student 36 (1968) note filed as erdos_1969_irrationality_certain_series, and Erdős and Straus, Pacific J. Math. 55 (1974), 85--92, are all called "On the irrationality of certain series". The 1957 note is the "[Er (57)]" that Erdős and Graham cite on printed p. 61 of their 1980 monograph for the $\varphi$ and $\sigma$ series. It does not contain the theorem that $\sum p_n^k/n!$ is irrational; the 1958 Enseignement Math. paper erdos_1958_sur_certaines_series_valeur_irrationnelle_french asserts that theorem for every $k$ and proves only the case $k=1$.

Contents

Throughout, $t>1$ is an integer, $d(n)$ is the number of divisors of $n$, $r(n)$ the number of solutions of $n=x^2+y^2$, $\varphi(n)$ Euler's function, $\sigma(n)$ the sum of the divisors, $\nu(n)$ the number of distinct prime factors and $\sigma_k(n)=\sum_{d\mid n}d^k$.

• Printed p. 212 recalls the 1948 theorem that $\sum d(n)/t^n$ and $\sum r(n)/t^n$ are irrational and restates, as unproved, the irrationality of $\sum\varphi(n)/t^n$, $\sum\sigma(n)/t^n$ and $\sum\nu(n)/t^n$ (remark on p. 212); the side remarks on $\sum 1/t^{n\pm\nu(n)}$, $\sum 1/t^{n\pm d(n)}$, $\sum 1/t^{n+\varphi(n)}$, $\sum 1/t^{n+\sigma(n)}$ and $\sum 1/t^{n+p_n}$ ($p_n$ the greatest prime factor of $n$) are recorded on that page.
• Printed p. 213 records the Erdős–Kac conjecture that $\sum\sigma_k(n)/n!$ is irrational for every integer $k>0$, with the cases $k=1,2$ proved (remark on p. 213), and asks whether $\sum 1/t^{n_k}$ can be algebraic when $\limsup n_k/k=\infty$.
• Theorem 1 (p. 213; proof pp. 213--215): $\sum_{n\ge1}1/t^{\varphi(n)}$ and $\sum_{n\ge1}1/t^{\sigma(n)}$ are irrational. The tools are Lemma 1, the irrationality criterion for series with bounded mean coefficients and sparse support, and Lemma 2 and Lemma 3 (pp. 214--215), which count how often $\varphi$ and $\sigma$ take values below $x$.
• Theorem 2 (p. 215; proof pp. 218--219): if $1<n_1<n_2<\cdots$ are integers with $\limsup n_k/k^l=\infty$, then $\sum_k1/t^{n_k}$ satisfies no algebraic equation with integer coefficients of degree at most $l$. Its tool is Lemma 4 (pp. 215--218), the criterion with signed coefficients of which Lemma 1 is a special case; the sharper Lemma 4′ is stated on p. 218 without proof.

Compiled scope

Every statement above was read on the page images. The proofs of Lemmas 2--4 and of Theorems 1--2 were read for their structure, which the result pages summarize with the paper's equation numbers and page pointers; no proof is rewritten in full and none has been independently reviewed. Nothing in the paper decides a catalog problem's status: for problems 249 and 250 it supplies restatements and exponent variants, for problem 252 a statement of the conjecture, and for problem 247 a partial result.

Bears on. #249, #250 and #69 (the p. 212 restatements; Theorem 1 is an exponent variant for #249 and #250), #252 (the p. 213 statement of the conjecture), #247 (the p. 213 question and Theorem 2).

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