Generalization of Pólya's Zero Distribution Theory for Exponential Polynomials, and Sharp Results for Asymptotic Growth

Abstract

An exponential polynomial of order q is an entire function of the form f(z)=P1(z)eQ1(z)+⋯+Pk(z)eQk(z), where the coefficients Pj(z),Qj(z) are polynomials in z such that max{deg(Qj)}=q.

In 1977 Steinmetz proved that the zeros of f lying outside of finitely many logarithmic strips around so called critical rays have exponent of convergence ≤q−1. This result does not say anything about the zero distribution of f in each individual logarithmic strip. Here, it is shown that the asymptotic growth of the non-integrated counting function of zeros of f is asymptotically comparable to rq in each logarithmic strip. The result generalizes the first order results by Pólya and Schwengeler from the 1920’s, and it shows, among other things, that the critical rays of f are precisely the Borel directions of order q of f. The error terms in the asymptotic equations for T(r, f) and N(r, 1/f) originally due to Steinmetz are also improved.