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

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.