On the ‘pits effect’ of Littlewood and Offord

Abstract

Asymptotic behaviour of the entire functions f(z)∑n=0 ∞ e2πinαnzn/n!, with real αn is studied. It turns out that the Phragmén-Lindelöf indicator of such a function is always non-negative, unless f(z)=eaz. For a special choice of αn=αn2 with irrational α, the indicator is constant and f has completely regular growth in the sense of Levin and Pfluger. Similar functions of arbitrary order are also considered. © 2007 London Mathematical Society.