Erdos 1948 arithmetical properties lambert series

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P. Erdős: On arithmetical properties of Lambert series, J. Indian Math. Soc. (N.S.) 12 (1948), 63--66 MR 10,594c; Zentralblatt 32,17.

For f(x) = sum_{n>=1} x^n/(1-x^n) = sum_{n>=1} tau(n) x^n and its sine analog g, Erdos states the single Theorem that f(1/t) and g(1/t) are irrational for every integer t with |t| > 1, extending Chowla's result for g at integers t >= 5 and confirming part of Chowla's conjecture. The proof is elementary and sieve-flavored: taking k = [(log n)^{1/10}] and the consecutive primes above (log n)^2, he solves a system of simultaneous congruences by the Chinese remainder theorem to produce, for arbitrarily large k, a block of consecutive integers whose divisor-function values force the base-t expansion of sum tau(r)/t^r to be non-terminating and non-periodic. Details are given only for f(1/t) with t > 1: the proof for g(1/t) is said to follow by this method and Chowla's (p. 63), and for negative t the paper names the extra step, that the expansion is not finite, says it "can be done by methods similar to those used above" and gives no details (p. 66). Vandehey (2012) completes the argument for f at negative t. The closing remark (p. 66, quoted below) flags the analogous questions for sum phi(n)/t^n, sum sigma(n)/t^n and sum vartheta(n)/t^n, vartheta(n) the number of prime factors of n, as hard. The paper is the reference point for problem 1049 (irrationality of sum 1/(t^n-1) = sum tau(n)/t^n for rational t > 1, still open beyond the integer case proved here), problem 1050 (sum 1/(2^n-3), a shifted variant later settled by Borwein), problem 257 (sum over an infinite set of 1/(2^n-1)), problem 258 (sum tau(n)/(a_1...a_n)) and problem 69 (sum omega(n)/2^n, one of the difficult analogs flagged at the end).

Closing remark, printed p. 66 (physical PDF p. 4), read on the page image: "The analogous problems about

$$\sum_{n=1}^{\infty}\frac{\phi(n)}{t^n},\qquad \sum_{n=1}^{\infty}\frac{\phi'(n)}{t^n},\qquad \sum_{n=1}^{\infty}\frac{\vartheta(n)}{t^n},$$

where $\phi(n)$ denotes Euler's $\phi$-function, $\phi'(n)$ denotes the sum of the divisors of $n$, and $\vartheta(n)$ denotes the number of prime factors of $n$, seem to present difficulties." The paper writes $\phi'$ for the sum of divisors and $\vartheta$ for the number of prime factors without saying whether repeated factors count; the restatement on p. 212 of erdos_1957_irrationality_certain_series reads the third function as the number of distinct prime factors. For the base $t=2$ these are the questions of problems 249, 250 and 69; the paper proves nothing about them.

Source: https://users.renyi.hu/~p_erdos/1948-04.pdf. No copyright or license line is printed on the file's four pages (pp. 63--66); 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/); the publisher's page could not be read (the journal's current site answered HTTP 403), and no Crossref record exists for the article, which has no DOI; the term is unstated.

Bears on. #69, #249, #250, #257, #258, #1049, #1050

Results to transcribe.

• Theorem: For every integer t with |t| > 1, both f(1/t) = sum_{n>=1} 1/(t^n-1) = sum_{n>=1} tau(n)/t^n and the sine analog g(1/t) are irrational.
• Method: A congruence construction with k = [(log n)^{1/10}], on the first k(k+1)/2 consecutive primes above (log n)^2 in blocks of 1, 2, ..., k (pp. 63--64), shows the base-t expansion of sum tau(r)/t^r cannot be finite or eventually periodic; the negative-t case needs the extra check that the expansion is infinite, which the paper says can be done by similar methods but does not carry out (p. 66).
• Closing remarks: Erdos regards the irrationality of sum phi(n)/t^n, sum sigma(n)/t^n and sum vartheta(n)/t^n, vartheta(n) the number of prime factors of n, as difficult and leaves all three open (p. 66).

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