On the irrationality of certain series: Problems and results. (English) Zbl 0656.10026

Erdős, Paul

New advances in transcendence theory, Proc. Symp., Durham/UK 1986, 102-109 (1988).

[For the entire collection see Zbl 0644.00005.]
The author presents a host of results and problems on the (ir)rationality of many interesting infinite series of rational numbers. For example: it is not known if \(\sum^{\infty}_{n=1}\omega(n)2^{-n}\) or \(\sum^{\infty}_{n=1}\phi (n)2^{-n}\) is irrational, where \(\omega(n)\) is the number of distinct prime divisors of \(n\) and \(\phi(n)\) is Euler’s function.
The paper also contains the proof of the following theorem. Let \(a_1<a_2<..\). be an infinite sequence of positive integers. Let \(c(n)=lcm(a_i| \quad a_i<n).\) Then, under certain hypotheses on the growth of the \(a_i\), the sum \(\sum^{\infty}_{n=1}c(n)^{-1}\) is irrational.

Reviewer: F.Beukers

MSC:

11J81	Transcendence (general theory)
00A07	Problem books

Keywords:

irrationality; rationality; problems; infinite series of rational numbers; number of distinct prime divisors; Euler’s function

Citations:

Zbl 0644.00005