On the zero sets of certain entire functions

Abstract

We consider the class B \mathbf B of entire functions of the form \[ f = ∑ p j exp ⁡ g j , f=\sum p_j\exp g_j, \] where p j p_j are polynomials and g j g_j are entire functions. We prove that the zero-set of such an f f , if infinite, cannot be contained in a ray. But for every region containing the positive ray there is an example of f ∈ B f\in \mathbf B with infinite zero-set which is contained in this region.