On several irrationality problems for Ahmes series

Abstract

Using basic tools of mathematical analysis and elementary probability theory we address several problems on the irrationality of series of distinct unit fractions, $\sum_k 1/a_k$. In particular, we study subseries of the Lambert series $\sum_k 1/(t^k-1)$ and two types of irrationality sequences $(a_k)$ introduced by Paul Erd\H{o}s and Ronald Graham. Next, we address a question of Erd\H{o}s, who asked how rapidly a sequence of positive integers $(a_k)$ can grow if both series $\sum_k 1/a_k$ and $\sum_k 1/(a_k+1)$ have rational sums. Our construction of double exponentially growing sequences $(a_k)$ with this property generalizes to any number $d$ of series $\sum_k 1/(a_k+j)$, $j=0,1,2,\ldots,d-1$, and, in particular, also gives a positive answer to a question of Erd\H{o}s and Ernst Straus on the interior of the set of $d$-tuples of their sums. Finally, we prove the existence of a sequence $(a_k)$ such that all well-defined sums $\sum_k 1/(a_k+t)$, $t\in\mathbb{Z}$, are rational numbers, giving a negative answer to a conjecture by Kenneth Stolarsky.