On several irrationality problems for Ahmes series

2024/06/25 by Kovač, Vjekoslav, Tao, Terence · 5 citations
#Classical Analysis and ODEs (math.CA) #FOS: Mathematics #Number Theory (math.NT)

paper · doi:10.48550/arxiv.2406.17593

Abstract

Using basic tools of mathematical analysis and elementary probability theory we address several problems on the irrationality of series of distinct unit fractions, ∑_k 1/a_k. In particular, we study subseries of the Lambert series ∑_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 ∑_k 1/a_k and ∑_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 ∑_k 1/(a_k+j), j=0,1,2,…,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 ∑_k 1/(a_k+t), t∈ℤ, are rational numbers, giving a negative answer to a conjecture by Kenneth Stolarsky.

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• Irrationality of rapidly converging series: A problem of Erdős and Graham
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