Davenport-Heilbronn Function Ratio Properties and Non-Trivial Zeros Study

Abstract

This paper systematically investigates the analytic properties of the ratio $f(s)/f(1-s)=X(s)$ based on the Davenport-Heilbronn functional equation $f(s)=X(s)f(1-s)$. We propose a novel method to analyze the distribution of non-trivial zeros through the monotonicity of the ratio $|f(s)/f(1-s)|$. Rigorously proving that non-trivial zeros can only lie on the critical line $\ddot{I}ƒ=1/2$, we highlight two groundbreaking findings: 1. Contradiction of Off-Critical Zeros: Numerical "exceptional zeros" (e.g., Spira, 1994) violate the theoretical threshold $\hat{I}º=1.21164$ and conflict with the monotonicity constraint of $|X(s)|=1$. 2. Essential Difference Between Approximate and Strict Zeros: Points satisfying $f(s)\to 0$ do not constitute strict zeros unless verified by analyticity. This work provides a new perspective for studying zero distributions of $L$-functions related to the Riemann Hypothesis.