On Certain Genus 0 Entire Functions

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 ${\left(-\frac{1}{x}\right)}^m\frac{d^k}{d{x}^k}\left(x^{k+m}\frac{d^m}{d{x}^m}\left(\frac{f^{\prime }(x)}{f(x)}\right)\right)$ 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.

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