On the "pits effect" of Littlewood and Offord

Alexandre Eremenko and Iossif Ostrovskii
Arxiv ID: 0710.2870Last updated: 12/21/2021
Suppose that the moduli of the coefficients of a power series are 1/n!, while the arguments are arbitrary. If an entire function f represented by such power series decreases exponentially on some ray, then it has to be an exponential. If the arguments of the coefficients are of the form 2pi n^2a, where a is irrational, then the function displays the so-called "pits effect". More precisely, under this condition, f is of completely regular growth with constant indicator.

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