Zentralblatt MATH

Publications of (and about) Paul Erdös

Zbl.No:  125.08602
Autor:  Erdös, Pál
Title:  On some applications of probability to analysis and number theory (In English)
Source:  J. Lond. Math. Soc. 39, 692-696 (1964).
Review:  The author discusses applications of probability theory for five problems of analysis, among them the following:
1. For what sequence of integers n1 < n2 < ··· does there exist a power series sumk = 1oo ak znk converging uniformly in |z| \leq 1 but for which sumk = 1oo |ak| = oo?
2. It is known that ft(z) = sumk = 0oo \epsilonkak zk where \epsilonk = ± 1, t = sumk = 1oo {1+\epsilonk \over 2k+1} and sumk = 1oo |ak|2 = oo, diverges almost everywhere on the unit circle if |ak| \geq ck where ck > 0 is a monotone sequence of numbers tending to zero so that

limsupk = oo [(sumj = 1k cj2)/ log (1/ck) ] > 0.

If this does not hold, is there a sequence {ak} such that |ak| \geq ck, for which ft(z) has at least one point of convergence for all t?
Some unpublished probabilistic methods in number theory conclude the paper.
Reviewer:  J.M.Gani
Classif.:  * 11N25 Distribution of integers with specified multiplicative constraints
                   11K99 Probabilistic theory
                   30B10 Power series (one complex variable)
Index Words:  probability theory

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