Davenport-Heilbronn Function Ratio Properties and Non-Trivial Zeros Study
Fuente:
arXiv
Saved in:
| Main Authors: | , |
|---|---|
| Format: | Preprint |
| Published: |
2025
|
| Subjects: | |
| Online Access: | |
| Tags: |
Add Tag
No Tags, Be the first to tag this record!
|
| _version_ | 1866908345521143808 |
|---|---|
| author | Liu, Tao Wu, Juhao |
| author_facet | Liu, Tao Wu, Juhao |
| contents | 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 $σ=1/2$, we highlight two groundbreaking findings: 1. Contradiction of Off-Critical Zeros: Numerical "exceptional zeros" (e.g., Spira, 1994) violate the theoretical threshold $κ=1.21164$ 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. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2503_24275 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Davenport-Heilbronn Function Ratio Properties and Non-Trivial Zeros Study Liu, Tao Wu, Juhao General Mathematics 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 $σ=1/2$, we highlight two groundbreaking findings: 1. Contradiction of Off-Critical Zeros: Numerical "exceptional zeros" (e.g., Spira, 1994) violate the theoretical threshold $κ=1.21164$ 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. |
| title | Davenport-Heilbronn Function Ratio Properties and Non-Trivial Zeros Study |
| topic | General Mathematics |
| url | https://arxiv.org/abs/2503.24275 |