Local geometric proof of Riemann Hypothesis

TL;DR

This paper provides a local geometric proof of the Riemann Hypothesis by analyzing the structure of the Riemann xi function and demonstrating its implications for the distribution of zeros.

Contribution

It introduces a local geometric approach to prove the Riemann Hypothesis by examining the peak-valley structure of the xi function in root-intervals.

Findings

Proves |u|>0 inside root-intervals for β>0

Shows v has opposite signs at interval endpoints

Establishes ||ξ||>0 for all t

Abstract

Riemann function $\xi(s)=u+iv, s=\beta+1/2+it$ has the important symmetry: $v=0$ if $\beta=0$. For $\beta>0$ we prove $|u|>0$ inside any root-interval $I_j=[t_j,t_{j+1}]$ and $v$ has opposite signs at two end-points of $I_j$. They imply local peak-valley structure and $||\xi||=|u|+|v/\beta|>0$ in $I_j$. Because each $t$ must lie in some $I_j$, then $||\xi||>0$ is valid for any $t$. By the equivalence $Re(\frac{\xi'}{\xi})>0$ of Lagarias(1999), we show that RH implies the peak-valley structure,which may be the geometric model expected by Bombieri(2000).