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**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 n_{1} < n_{2} < ··· does there exist a power series **sum**_{k = 1}^{oo} a_{k} z^{nk} converging uniformly in |z| \leq 1 but for which **sum**_{k = 1}^{oo} |a_{k}| = oo?

2. It is known that f_{t}(z) = **sum**_{k = 0}^{oo} \epsilon_{k}a_{k} z^{k} where \epsilon_{k} = ± 1, t = **sum**_{k = 1}^{oo} {1+\epsilon_{k} \over 2^{k+1}} and **sum**_{k = 1}^{oo} |a_{k}|^{2} = oo, diverges almost everywhere on the unit circle if |a_{k}| \geq c_{k} where c_{k} > 0 is a monotone sequence of numbers tending to zero so that **limsup**_{k = oo} **[****(****sum**_{j = 1}^{k} c_{j}^{2}**)**/ log (1/c_{k}) **]** > 0. If this does not hold, is there a sequence **{**a_{k}**}** such that |a_{k}| \geq c_{k}, for which f_{t}(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Some unpublished probabilistic methods in number theory conclude the paper.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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