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On Certain Genus 0 Entire Functions
Published 8 Jun 2022 in math.CA | (2206.05104v8)
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.
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