论文标题
在某些属0中，整个功能
On Certain Genus 0 Entire Functions
论文作者
论文摘要
在这项工作中，我们证明整个函数$ f（z）$仅在且仅当其订单严格少$ 1 $时才具有负零，其根部序列是真正的主导，并且存在一个非负整数$ m $ $ \ 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）$对于所有非负整数$ k $的$（0，\ infty）$，完全单调。作为一种应用，我们陈述了Riemann假设的必要条件，并为原始Dirichlet特征提供了普遍的Riemann假设。
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.
