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Source. Printed p. 212, physical PDF p. 1, first paragraph, read on the page image.

Statement

The passage reads, with $t>1$ an integer: "In my above paper I remarked that I cannot prove that any of the series

$$\sum_{n=1}^{\infty}\frac{\varphi(n)}{t^n},\qquad \sum_{n=1}^{\infty}\frac{\sigma(n)}{t^n},\qquad \sum_{n=1}^{\infty}\frac{\nu(n)}{t^n}$$

are irrational, where $\varphi(n)$ is Euler's $\varphi$ function, $\sigma(n)$ the sum of the divisors of $n$ and $\nu(n)$ the number of distinct prime factors of $n$." The "above paper" is footnote 1, "Indian Journal of Math. 12, 63--66 (1948)", the Lambert-series paper filed as erdos_1948_arithmetical_properties_lambert_series, whose closing remark on p. 66 first posed the three questions.

This is a statement of what the paper does not prove; no result in it concerns these three series. The rest of the paragraph records: that $\sum 1/t^{n+\nu(n)}$ and $\sum 1/t^{n-\nu(n)}$ can be proved irrational "by the methods used in the above paper" (no proof is given); that the author failed for $\sum 1/t^{n+d(n)}$ and $\sum 1/t^{n-d(n)}$ because he cannot show that the paper's (1),

$$\max_{m\le n}\bigl(m+d(m)\bigr)<\min_{m>n}\bigl(m+d(m)\bigr),$$

holds for infinitely many $n$ (footnote 2: with $\nu(m)$ in place of $d(m)$ this "is essentially contained in" the 1948 paper); and that he "cannot prove anything about" $\sum 1/t^{n+\varphi(n)}$, $\sum 1/t^{n+\sigma(n)}$ and $\sum 1/t^{n+p_n}$, $p_n$ the greatest prime factor of $n$, since the analogue of (1) with $\varphi$, $\sigma$ or the greatest prime factor in place of $d$ is false.

Relation to the catalog

Problem 249 asks the first question for $t=2$, Problem 250 the second and Problem 69 the third ($\nu=\omega$). The paper poses them for every integer base $t>1$; the catalog fixes $t=2$, and the all-base statement is a variant of each problem, not the problem. Theorem 1 of the same paper settles the exponent variants $\sum 1/t^{\varphi(n)}$ and $\sum 1/t^{\sigma(n)}$, which are different series and say nothing about Problems 249 and 250.

Bears on. #249, #250 and #69, as a 1957 restatement of each question; the paper records no progress on any of them.