Entire functions with an arithmetic sequence of exponents

TL;DR

This paper investigates the zero distribution of entire functions with exponents in arithmetic sequences, establishing conditions for zeros to lie on specific radial rays in the complex plane.

Contribution

It provides new criteria for the zero distribution of functions with exponents in arithmetic progressions, extending understanding of their geometric zero patterns.

Findings

Zeros of the functions lie on $m$ radial rays under certain conditions.

Conditions relate to the coefficients and the structure of the exponents.

The results describe the geometric arrangement of zeros in the complex plane.

Abstract

For a given entire function $f(z)=\sum_{j=0}^{\infty}a_{j}z^{j}$, we study the zero distribution of $f_{r}(z)=\sum_{j\equiv r\pmod m}a_{j}z^{j}$ where $m\in\mathbb{N}$ and $0\le r<m$. We find conditions under which the zeros of $f_{r}(z)$ lie on $m$ radial rays defined by $\Im z^{m}=0$ and $\Re z^{m}\le0$.