Christian Lawson-Perfect @christianp

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Vjeko Kovac and I just uploaded a preprint to the arXiv entitled "On several irrationality problems for Ahmes series" https://arxiv.org/abs/2406.17593 , making progress on several open problems of Erdos and of Stolarsky regarding when sums of reciprocals of natural numbers are rational or not. Perhaps the most striking result is that we can find an increasing sequence 𝑎ₙ of natural numbers with the property that ∑ₙ1/(𝑎ₙ+𝑡) converges to a rational for all rational 𝑡 for which the denominators are non-zero, answering a question of Stolarsky in the negative. More discussion at https://terrytao.wordpress.com/2024/11/27/on-several-irrationality-problems-for-ahmes-series/

arXiv.orgOn several irrationality problems for Ahmes seriesUsing 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 Erdős and Ronald Graham. Next, we address a question of Erdős, 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 Erdős 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.

Nov 28, 2024, 06:25 AM··Web

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