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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 $n$ (p. 638).

Lemma 2.14 (p. 640). "If there exists a positive constant $c$ so that $|a_n|>c(\log n)^{3/4}$ for all $n$ then the series (2.1) is irrational."

The paper adds (p. 640) that in this lemma "we need not assume the monotonicity of $a_n$", nor even that the $a_n$ are positive, and that the proof is given for positive $a_n$ only. So the section's convention $2\le a_1\le a_2\le\cdots$ is not a hypothesis of the lemma; the proof as printed covers positive integers $a_n$.

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

Read depth. Claims checked: the statement and the remark on monotonicity were read on the page image of p. 640; the proof was read for structure, not checked. Nothing here is independently reviewed.

Proof pointer

Pages 640--641. The proof combines Lemma 2.17 with the almost-all bound $d(n)<(\log n)^{\log2+\varepsilon}$ (2.16) to find infinitely many $n$ at which every later value $d(n+y)$ is small compared with $c^y(\log n)^{3y/4}$. If $\xi=a/b$, multiplying by $a_1\cdots a_n b$ leaves an integer plus a tail $b\sum_{y\ge1}d(n+y)/(a_{n+1}\cdots a_{n+y})$ that lies strictly between $0$ and $1$ (2.22), a contradiction.

Dependencies

Lemma 2.17 (pp. 640--641); the Dirichlet divisor theorem $\sum_{n\le N}d(n)\sim N\log N$ (2.15); the bound (2.16), which the paper attributes to M. Kac, Note on the distribution of values of the arithmetic function $d(m)$, Bull. Amer. Math. Soc. 47 (1941), 815--817 (its reference [3]).

Bears on

• #258: covers every sequence of positive integers with $a_n>c(\log n)^{3/4}$ for all $n$ and some constant $c>0$, monotone or not; it says nothing about sequences tending to infinity more slowly or irregularly.
• #252: $a_n=n$ satisfies the hypothesis with $c=1$ (at $n=1$ the bound is $0$), so $\sum d(n)/n!$ is irrational. This is the divisor-count case $k=0$, outside the problem's range $k\ge1$; the paper does not state the $n!$ case.