Constraint Structure and Zero Counting in the Integral Representation of the Zeta Function

TL;DR

This paper reinterprets Riemann's integral representation of $\zeta(s)$ through a geometric framework that leverages the functional equation to decompose the analytic structure into a real-part projection on the critical line. It introduces a phase function $\omega(t)=\arg\varphi(\tfrac12+it)$ so that zeros on the critical line correspond to phase crossings $\omega(t)=(k+\tfrac12)\pi$, turning a complex-analytic problem into a one-dimensional geometric counting task. The authors derive an exact zero-count relation $N(T)=\frac{\theta(T)+\arg\zeta(\tfrac12+iT)}{\pi}+1$ and prove, within this framework, that off-line zeros must vanish ($N_r(T)=0$) while revealing a tight coherence between global boundary variation and local critical-line behavior. Overall, the work presents a non-asymptotic, structurally determined approach to zero counting that emphasizes geometric rigidity and the intrinsic connection between global topology and local sign oscillations in analytic number theory.

Abstract

Starting from the classical integral representation of the $ζ(s)$ function introduced by Riemann in 1859, this paper reexamines its analytic symmetry structure. By performing a geometric decomposition of the integral representation, we demonstrate that on the critical line $\Re(s)=\frac{1}{2}$, the value of $ξ(s)$ corresponds strictly to the \textbf{real-part projection} of a specific analytic component. This discovery equivalently transforms the problem of complex zeros into a problem of \textbf{sign evolution} along the real axis. Based on this geometric framework, we \textbf{construct} an analytic mechanism of \textbf{"Two-End Anchoring, Interval Counting"}: the global argument increment on the region boundary \textbf{anchors} the initial value of the phase function, while the geometric decomposition structure on the critical line \textbf{locks} its final value. This mechanism reveals an \textbf{intrinsic coherence} between global topological constraints and local sign oscillations. Unlike traditional methods that rely on asymptotic estimates (such as the Big $O$ error term), the analysis in this paper is \textbf{grounded in} exact identities. It unveils the \textbf{geometric determinism} underlying the zero-counting formula, offering a novel perspective for analytic number theory independent of asymptotic analysis.

Figures (1)

• Figure 1: Four possible geometric configurations of equation \ref{['implicit_sym']} on the critical line $s=1/2+iT$, demonstrating the real-part projection mechanism induced by the functional equation.

Theorems & Definitions (5)

• Theorem 4.1: Global Geometric Constraint Theorem
• Theorem 4.2: Phase Locking Theorem
• proof
• proof
• Remark 5.1