A few properties of the ratio of Davenport-Heilbronn Functions

Abstract

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 $\{s_n|\sigma \ne 1/2\}$ presents two puzzles: 1) $f(s_n)\ne f(1-s_n)$; 2) the existence of non-trivial zeros $\{s_n|\sigma \ne 1/2\}$ is in contradiction of the monotonicity of the similar ratio $\left|\frac{f(s)}{f(1-s)}\right|$.

8 pages, 1 figure