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 $\left(-\frac{1}{x}\right)^{m}\frac{d^{k}}{dx^{k}}\left(x^{k+m}\frac{d^{m}}{dx^{m}}\left(\frac{f'(x)}{f(x)}\right)\right)$ 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.
- Publication:
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arXiv e-prints
- Pub Date:
- June 2022
- DOI:
- 10.48550/arXiv.2206.05104
- arXiv:
- arXiv:2206.05104
- Bibcode:
- 2022arXiv220605104Z
- Keywords:
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- Mathematics - Classical Analysis and ODEs
- E-Print:
- 13pages