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**Zentralblatt MATH**

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

**Zbl.No: ** 256.30025

**Autor: ** Erdös, Paul; Renyi, Alfréd

**Title: ** On random entire functions. (In English)

**Source: ** Zastosowania Mat. 10, 47-55 (1969).

**Review: ** Let f(z) = **sum** ^{oo}_{n = 0}a_{n}z^{n} be an arbitrary entire function, held fixed in all that follows. For r > 0 let M(r) = **max** (|f(z)|: |z| = r) be the maximum modulus function of f and \mu(r) = **max** (|a_{n}| r^{n}: n \geq 0) the maximum term in the series expansion of f. The following extension of Wiman's theorem was proved by Rosenbloom: for every \delta > 0 there exists a subset E_{\delta} of fintie logarithmic measure such that if r \not in E_{\delta}, then M(r) < \mu (r)[ log \mu (r)]^{ ½}[ log log \mu (r)]^{1+\delta}. For 0 \leq t < 1 let R_{n}(t) = sign \sin (2^{n} \pi t) denote the n-th Rademacher function, n \geq 0. The present paper considers the class of entire functions obtained by giving random signs to the terms in the series expansion of f above; explicitly, the entire functions which can be written f(z,t) = **sum** ^{oo}_{n = 0} a_{n}R_{n}(t)z^{n}, 0 \leq t < 1. Keeping the notation above, let M(r,t) = **max** (|f(z,t)|: |z| = r). The main result is: for every \delta > 0 and almost all t in [0,1), there exists a subset E_{\delta} (t) \subset R_+ of finite logarithmic measure (depending on t) such that for r \not in E_{\delta} (t),

M(r,t) < \mu (r)[ log \mu (r)]^{1/4}[ log log \mu (r)]^{1+\delta}. Two related results are also given.

**Reviewer: ** T.P.Speed

**Classif.: ** * 30D20 General theory of entire functions

60-XX Probability theory and stochastic processes

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