An approximate solution to Erd\"os' maximum modulus points problem

Abstract

In this note we investigate the asymptotic behavior of the number of maximum modulus points, of an entire function, sitting in a disc of radius $r$. In 1964, Erd\Humlaut{o}s asked whether there exists a non-monomial function so that this quantity is unbounded? tends to infinity? In 1968 Herzog and Piranian constructed an entire map for which it is unbounded. Nevertheless, it is still unknown today whether it is possible that it tends to infinity or not. In this paper, we construct a transcendental entire function that is arbitrarily close to satisfying this property, thereby giving the strongest evidence supporting a positive answer to this question.