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Source. Theorem 5.2, preprint p. 10; proof p. 10; Remark and its example pp. 10--11; Example 5.1, p. 11; the introduction, p. 2. Read on the rendered pages.

Statement

Theorem 5.2 (p. 10): "Let $\{a_n\}_{n=1}^{\infty}$ and $\{b_n\}_{n=1}^{\infty}$ be two monotonic sequences of positive integers such that $\lim_{n\to\infty}\frac{b_n}{a_n^2}=0$ and $\lim_{n\to\infty}\frac{a_{2n}b_{2n}}{n\cdot a_n^2}=0$. Suppose for every $\epsilon>0$ there exists $\delta>0$ and infinitely many $N$ such that $\gcd(a_n-1,b_n)<\epsilon b_n$ for at least $\delta N$ integers $n\in[N,2N)$. Then $S=\sum_{n=1}^{\infty}\frac{b_n}{a_1\ldots a_n}$ is irrational."

The paper introduces it (p. 10) as an instance of relaxing the primality of $p_n$ in Theorem 5.1. The introduction (p. 2) draws an exact test from it: if $a_n$ and $b_n$ are monotonic sequences of positive integers with $b_n=o(a_n^2)$, $\gcd(a_n-1,b_n)=1$ for all $n$ and $a_{2n}b_{2n}=o(na_n^2)$, then $S$ is rational if and only if $b_n/(a_n-1)$ is constant for $n$ greater than some $n_0$.

Proof sketch (p. 10)

If $S=r/q$, the tails $S_n$ of (2) satisfy $qS_n\in\mathbb Z$ (Lemma 2.1) and stay within $\epsilon$ of $b_n/a_n$ by (3). An equality $S_{n+1}=S_n$ means $(a_n-1)S_n=b_n$, so $a_n-1$ divides $qb_n$; the gcd hypothesis with $\epsilon=1/q$ rules this out for a positive proportion of $n\in[N,2N)$, for infinitely many $N$. A strict rise or fall of $S_n$ moves $b_n/a_n$ by nearly $1/q$, which forces $b_n$ or $a_n$ to grow by a definite amount; counting such steps in $[N,2N)$ bounds their number by a constant times $a_{2N}b_{2N}/a_N^2$, a vanishing proportion of $N$ by the second limit. The two counts contradict each other.

The growth condition cannot be dropped (pp. 10--11)

The Remark after the proof says that the condition $a_{2N}b_{2N}=o(Na_N^2)$ may be relaxed by the argument used for the case $S_n>S_{n+1}$ in the proof of Theorem 5.1, but not simply dropped. Its example: $a_1=10$, $b_1=9$, $a_2=13$, $b_2=11$ and, recursively,

$$a_{2n-1}=a_{2n-2}+2,\quad b_{2n-1}=2a_{2n-1}-1,\quad b_{2n}=b_{2n-1}+2,\quad a_{2n}=b_{2n}+2 .$$

Both sequences increase, $\gcd(a_n-1,b_n)=\gcd(a_n,b_n)=1$ for every $n$, $b_n/a_n$ has only the limit points $1$ and $2$, and the partial sums are $1-1/(a_1\cdots a_N)$ for odd $N$ and $1-2/(a_1\cdots a_N)$ for even $N$, so $S=1$.

Example 5.1 (p. 11)

Theorem 5.2 shows that $\sum_{n\ge1}(p_n/n!)^k$ is irrational for every integer $k\ge1$. This series differs from $\sum p_n^k/n!$, the series of Erdős's 1958 claim.

Bears on. No catalog problem directly.