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 sumoon = 1\omega(n)2-n or sumoon = 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 a1 < a2 < ... be an infinite sequence of positive integers. Let c(n) = lcm(ai|   ai < n). Then, under certain hypotheses on the growth of the ai, the sum sumoon = 1c(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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