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

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

Zbl 0644.00005

### Online Encyclopedia of Integer Sequences:

Decimal expansion of Sum_{k>=2} 1/(k! - 1).