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Statement

Setting (p. 222). The $n_i$ are the terms of an infinite sequence $n_1<n_2<\cdots$ of positive integers, the setting of the series (2) on the same page, and $t$ is an integer. The hypothesis $(n_i,n_j)=1$ is printed without a range; it is meant for $i\ne j$, as the proof says the $n$'s are relatively prime in pairs (p. 223).

Theorem (p. 222, unnumbered, quoted). "Let $(n_i, n_j)=1$, $\sum_{i=1}^{\infty}1/n_i\leqslant\infty$ [sic]. Then $\sum_{i=1}^{\infty}\frac{1}{t^{n_i}-1}$ is irrational for every $t\geqslant2$."

The printed $\leqslant\infty$ is a misprint for $<\infty$: the paragraph that follows on p. 222 calls the hypothesis "the condition $\sum_{i=1}^{\infty}1/n_i<\infty$", and the proof uses the convergence of $\sum_j1/n_j$ (p. 224). In the corpus's words: if the $n_i$ are pairwise coprime and $\sum_i1/n_i<\infty$, then

$$\sum_{i=1}^{\infty}\frac{1}{t^{n_i}-1}$$

is irrational for every integer $t\ge2$.

Remarks in the paper. Erdős states without details that more complicated arguments show the coprimality condition to be superfluous, that the convergence condition could be replaced by a weaker but more complicated one, and that he expects the series to be irrational whenever $n_{k+1}-n_k\to\infty$, perhaps even whenever $n_k/k\to\infty$ (p. 222). On p. 226 he adds that without coprimality the proof needs the fact that $\alpha$ is irrational when the fractional parts of $t^n\alpha$ take infinitely many values, and that with coprimality Brun's method could probably weaken the convergence condition to $\sum_{n_i<x}1/n_i=o(\log\log x)$, but that he does not see how to treat the case where the $n_i$ are all the primes. None of these extensions is proved in the paper.

Source. P. Erdős, On the irrationality of certain series, Math. Student 36 (1968), 222--226 (1969): the Theorem on p. 222, its proof on pp. 223--225. The edition read is identified on the source card.

Read depth. Claims checked: the setting and the statement were read clause by clause on the printed page. The proof was read but not checked step by step. Nothing here is independently reviewed.

Proof pointer

Pp. 223--225. Writing $V^*(m)$ for the number of the $n_i$ that divide $m$, the sum equals $\alpha=\sum_{m\ge1}V^*(m)/t^m$ (equation (3), p. 223). The proof shows that the base-$t$ expansion of $\alpha$ does not terminate yet has, for every large $k$, a run of at least $k/2$ zero digits. For this it chooses $y$ through $k$ simultaneous congruences modulo products of the $n_i$, which pairwise coprimality makes consistent, so that $V^*(y+i)=t^i$ for $1\le i\le k$, and then bounds the tail $\sum_{j>k}V^*(y+j)/t^{y+j}$ for most admissible $y$ (estimates (5) to (14), with an unnumbered Lemma on p. 225 bounding $V^*(y+j)$ by $j^2$ for all but a small proportion of the $y$). The method follows Erdős's earlier paper on Lambert series (J. Indian Math. Soc. 12 (1948), 63--66), the paper's reference [1].

Dependencies

None in this corpus.

Bears on

• Problem 257: at $t=2$, every infinite pairwise coprime $A$ with $\sum_{a\in A}1/a<\infty$ is an instance of the problem, and the theorem answers it yes; it says nothing about supports in which two members share a factor or whose reciprocal sum diverges. The problem's claim page records this class.