论文标题
Davenport-Heilbronn函数比率的一些属性
A few properties of the ratio of Davenport-Heilbronn Functions
论文作者
论文摘要
从davenport-heilbronn函数方程式开始：$ f（s）= x（s）f（1-s）$，我们发现了Meromororphic函数的四个属性$ x（s）$定义为Davenport-Heilbronn函数的比率：$ \ frac {f（s）} {f（1-s）} {f（1-s）{lem和三个$ s $ s）。我们首次提议研究Davenport-Heilbronn函数的非平凡零的分布，探索相似性比$ \ left |的单调性。 \ frac {f（s）} {f（1-s）} \ right | $。我们指出的是，对于满足Davenport-Heilbronn函数方程的$ f（s）$，关键行$ \ {s_n |存在非平凡零的存在。 σ\ neq 1/2 \} $提出了两个难题：1）$ f（s_n）\ neq f（1 -s_n）$; 2）非平凡零的存在$ \ {s_n | σ\ neq 1/2 \} $与相似比率$ \ left |的单调性相矛盾\ frac {f（s）} {f（1-s）} \ right | $。
Starting from the Davenport-Heilbronn function equation: $f(s) = X(s) f(1-s)$, we discover the four properties of the meromorphic function $X(s)$ defined as the ratio of the Davenport-Heilbronn functions: $\frac{f(s)}{f(1-s)} = X(s)$, and three corresponding lemmas. For the first time, we propose to study the distribution of the non-trivial zeros of the Davenport-Heilbronn function by exploring the monotonicity of the similarity ratio $\left| \frac{f(s)}{f(1-s)} \right|$. We point out that for the $f(s)$ which satisfies the Davenport-Heilbronn function equation, the existence of non-trivial zeros outside of the critical line $\{ s_n | σ\neq 1/2 \}$ presents two puzzles: 1) $f(s_n) \neq f(1 - s_n)$; 2) the existence of non-trivial zeros $\{ s_n | σ\neq 1/2 \}$ is in contradiction of the monotonicity of the similar ratio $\left| \frac{f(s)}{f(1-s)}\right|$.
