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Statement

The series is the paper's (2.1),

$$\xi=\sum_{n=1}^{\infty}\frac{d(n)}{a_1a_2\cdots a_n},$$

with $d(n)$ the number of divisors of $n$ and the $a_n$ positive integers (p. 638).

Theorem 2.23 (p. 641). "The series (2.1) is irrational whenever

$$2\leq a_1\leq a_2\leq\cdots\leq a_n\leq\cdots\text{."}$$

The sequence need not tend to infinity: a bounded nondecreasing sequence is eventually constant, and it is covered. Some restriction is needed: the paper notes (p. 638) that $a_n=d(n)+1$ gives $\xi=1$. It adds (p. 641), without proof, that with considerable additional effort the monotonicity can be weakened to $a_m/a_n\ge c>0$ for all $m>n$; that stronger form is not proved in the paper.

Source. P. Erdős, E. G. Straus, Some number theoretic results, Pacific J. Math. 36 (1971), no. 3, 635--646; Theorem 2.23 on p. 641, assembled from Lemma 2.2 (pp. 638--640) and Lemma 2.14 (pp. 640--641). The copy read is identified on the source card.

Read depth. Claims checked: the statement was read on the page image of p. 641 and the two lemmas on the page images of pp. 638 and 640; their proofs were read for structure, not checked. Nothing here is independently reviewed.

Proof pointer

The paper says only "Summing up we have" (p. 641); the joining is spelled out here. If some $\delta>0$ has $a_n<(\log n)^{1-\delta}$ for infinitely many $n$, Lemma 2.2 applies. Otherwise, taking $\delta=1/8$, $a_n\ge(\log n)^{7/8}$ for all large $n$, which exceeds $(\log n)^{3/4}$; since every $a_n\ge2$, a small enough constant $c>0$ gives $a_n>c(\log n)^{3/4}$ for all $n$, and Lemma 2.14 applies.

Dependencies

Lemma 2.2 and Lemma 2.14, the latter through Lemma 2.17. It generalizes P. Erdős, On arithmetical properties of Lambert series, J. Indian Math. Soc. 12 (1948), 63--66, the case $a_n=t$ constant (p. 638).

Bears on

• #258: proves the problem's irrationality for every nondecreasing sequence of integers with $a_1\ge2$. A nondecreasing sequence of positive integers that begins with some $1$s but is not constantly $1$ reduces to this case by replacing its $j$ leading $1$s with $2$s: the new series is a rational number plus $2^{-j}$ times the old one, so the two are irrational together; this reduction is this page's. Sequences tending to infinity without being monotone are outside the theorem; the problem's question for them is the paper's Conjecture 2.24.
• #252: the sequence $2,2,3,4,5,\ldots$ satisfies the hypothesis and its series is $\tfrac12\sum d(n)/n!$, so $\sum d(n)/n!$ is irrational: the divisor-count case $k=0$, outside the problem's range $k\ge1$. The reduction is this page's; the paper does not state the $n!$ case.