Generalization of P\'olya's zero distribution theory for exponential polynomials, plus sharp results for asymptotic growth

TL;DR

This paper extends Pólya's zero distribution theory for exponential polynomials, providing detailed asymptotic growth results for zeros in each logarithmic strip and refining earlier estimates on zero distribution and Borel directions.

Contribution

It generalizes Pólya's and Schwengeler's results by analyzing zero distribution in individual logarithmic strips and improves asymptotic estimates for exponential polynomials.

Findings

Zeros of exponential polynomials grow like r^q in each logarithmic strip.

Critical rays are exactly the Borel directions of order q.

Refined asymptotic formulas for zero counting functions.

Abstract

An exponential polynomial of order $q$ is an entire function of the form $$ f(z)=P_1(z)e^{Q_1(z)}+\cdots +P_k(z)e^{Q_k(z)}, $$ where the coefficients $P_j(z),Q_j(z)$ are polynomials in $z$ such that $$ \max\{\deg{Q_j}\}=q. $$ 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 $\leq q-1$. This result does not say nothing 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 $r^q$ in each logarithmic strip. The result generalizes the first order results by P\'olya 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.