On Certain Genus 0 Entire Functions

TL;DR

This paper characterizes entire functions with only negative zeros based on their order, root distribution, and certain monotonicity conditions, and applies these results to conditions related to the Riemann hypothesis.

Contribution

It provides a new characterization of genus 0 entire functions with negative zeros and links these properties to the Riemann hypothesis and its generalizations.

Findings

Entire functions with only negative zeros have order less than 1.

A specific monotonicity condition characterizes these functions.

Application to necessary and sufficient conditions for the Riemann hypothesis.

Abstract

In this work we prove that an entire function $f(z)$ has only negative zeros if and only if its order is strictly less $1$, its root sequence is real-part dominating and there exists an nonnegative integer $m$ the real function $\left(-\frac{1}{x}\right)^{m}\frac{d^{k}}{dx^{k}}\left(x^{k+m}\frac{d^{m}}{dx^{m}}\left(\frac{f'(x)}{f(x)}\right)\right)$ are completely monotonic on $(0,\infty)$ for all nonnegative integer $k$. As an application we state a necessary and sufficient condition for the Riemann hypothesis and generalized Riemann hypothesis for a primitive Dirichlet character.