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**Zentralblatt MATH**

**Publications of (and about) Paul Erdös**

**Zbl.No: ** 656.10026

**Autor: ** Erdös, Paul

**Title: ** On the irrationality of certain series: Problems and results. (In English)

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

**Review: ** [For the entire collection see Zbl 644.00005.]

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**^{oo}_{n = 1}\omega(n)2^{-n} or **sum**^{oo}_{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}| a_{i} < n). Then, under certain hypotheses on the growth of the a_{i}, the sum **sum**^{oo}_{n = 1}c(n)^{-1} is irrational.

**Reviewer: ** F.Beukers

**Classif.: ** * 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 644.00005

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