Computational Methods and Function Theory ( IF 0.6 ) Pub Date : 2020-07-30 , DOI: 10.1007/s40315-020-00336-7 Janne M. Heittokangas , Zhi-Tao Wen
An exponential polynomial of order q is an entire function of the form
$$\begin{aligned} f(z)=P_1(z)e^{Q_1(z)}+\cdots +P_k(z)e^{Q_k(z)}, \end{aligned}$$where the coefficients \(P_j(z),Q_j(z)\) are polynomials in z such that
$$\begin{aligned} \max \{\deg (Q_j)\}=q. \end{aligned}$$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 \(\le q-1\). 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 \(r^q\) 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.
中文翻译：
指数多项式的Pólya零分布理论的推广和渐近增长的尖锐结果
q阶的指数多项式是形式的完整函数
$$ \ begin {aligned} f（z）= P_1（z）e ^ {Q_1（z）} + \ cdots + P_k（z）e ^ {Q_k（z）}，\ end {aligned} $$其中系数\（P_j（z），Q_j（z）\）是z中的多项式，使得
$$ \ begin {aligned} \ max \ {\ deg（Q_j）\} = q。\ end {aligned} $$1977年Steinmetz证明，位于所谓的临界射线周围有限对数条带外的f的零点具有收敛指数\（\ le q-1 \）。这个结果没有说出每个对数带中f的零分布。在这里，表明在每个对数带中，f的零的非积分计数函数的渐近增长与\（r ^ q \）渐近可比。该结果概括了Pólya和Schwengeler从1920年代起的一阶结果，并且除其他外，它表明f的临界射线恰好是f的q阶的Borel方向。。最初由于Steinmetz引起的T（r， f）和N（r，1 / f）渐近方程中的误差项也得到了改善。
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