Bank-Laine functions with real zeros

TL;DR

This paper investigates real Bank-Laine functions with unbounded real zeros, establishing a dichotomy: such functions either have explicit trigonometric forms or their zeros have a high exponent of convergence, with an example confirming the result's sharpness.

Contribution

It provides a classification of finite order real Bank-Laine functions with unbounded real zeros, showing a clear dichotomy and constructing an example to demonstrate the sharpness of the theorem.

Findings

Functions either have explicit trigonometric representations or zeros with exponent of convergence at least 3.

Constructed example via quasiconformal surgery confirms the sharpness of the main result.

Zeros are all real but unbounded, with finite order of the functions.

Abstract

Every real Bank-Laine function of finite order, whose zeros are all real but neither bounded above nor bounded below, either has an explicit representation in terms of trigonometric functions or has zeros with exponent of convergence at least 3. An example constructed via quasiconformal surgery demonstrates the sharpness of this result.