On the genus of meromorphic functions

TL;DR

This paper introduces the class of LLD meromorphic functions, explores their properties related to Weierstrass genus, and explains why many classical functions have minimal genus based on these new concepts.

Contribution

It defines LLD meromorphic functions, introduces their vertical order and convergence exponent, and proves conditions under which their Weierstrass genus is minimal, clarifying classical function behaviors.

Findings

Many classical functions have minimal Weierstrass genus.

The conditions $m_0(f) \,\leq\, d(f)$ ensure minimal genus.

The concepts of vertical order and convergence exponent are key to understanding genus.

Abstract

We define the class of Left Located Divisor (LLD) meromorphic functions and their vertical order $m_0(f)$ and their convergence exponent $d(f)$. When $m_0(f)\leq d(f)$ we prove that their Weierstrass genus is minimal. This explains the phenomena that many classical functions have minimal Weierstrass genus, for example Dirichlet series, the $\Gamma$-function, and trigonometric functions.