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

Tao Liu; Juhao Wu

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arXiv · 2503.24275

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

Tao Liu·Juhao Wu

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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 $\sigma=1/2$, we highlight two groundbreaking findings: 1. Contradiction of Off-Critical Zeros: Numerical "exceptional zeros" (e.g., Spira, 1994) violate the theoretical threshold $\kappa=1.21164$ 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.

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BibTeX RIS

Tao Liu, Juhao Wu. 2025-03-31. Davenport-Heilbronn Function Ratio Properties and Non-Trivial Zeros Study. https://arxiv.org/abs/2503.24275

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