A few properties of the ratio of Davenport-Heilbronn Functions

TL;DR

This paper investigates properties of the ratio of Davenport-Heilbronn functions, exploring the distribution of their zeros and proposing a new approach based on the monotonicity of the ratio to understand their zero locations.

Contribution

It introduces four properties of the ratio of Davenport-Heilbronn functions and proposes studying zero distribution via the monotonicity of this ratio, a novel approach.

Findings

Identified four key properties of the ratio X(s)

Proposed the monotonicity of |f(s)/f(1-s)| as a tool for zero distribution analysis

Highlighted contradictions in zero locations outside the critical line

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