Constraint Structure and Zero Counting in the Integral Representation of the Zeta Function
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 (s)=12, the value of (s) corresponds strictly to the real-part projection of a specific analytic component. This discovery equivalently transforms the problem of complex zeros into a problem of sign evolution along the real axis. Based on this geometric framework, we construct an analytic mechanism of "Two-End Anchoring, Interval Counting": the global argument increment on the region boundary anchors the initial value of the phase function, while the geometric decomposition structure on the critical line locks its final value. This mechanism reveals an 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 grounded in exact identities. It unveils the geometric determinism underlying the zero-counting formula, offering a novel perspective for analytic number theory independent of asymptotic analysis.