What question did this study set out to answer?

The research aims to resolve the conflict between discrete arithmetic spectra and continuous modular spectra.

May 2, 2026Open Access

Arithmetic Spectral Flow and the Rigidity of Zeros of the Riemann Zeta Function

Key Points

The research aims to resolve the conflict between discrete arithmetic spectra and continuous modular spectra.
Introduced a dynamic evolution model to establish arithmetic spectral flow.
Constructed time-dependent spectral measures and verified equations using mollification approximation and Picard iteration.
Conducted analyses on spectral free energy and global non-blowup estimations.
Proved that all non-trivial zeros of the Riemann zeta function lie on the critical line Re(s)=1/2.
Derived a global uniform spectral gap theorem.
Distinguished finite-time collisions from infinite asymptotic behavior of zeros.

Abstract

This paper abandons the traditional static operator construction framework and adopts a dynamic evolution paradigm to establish the theory of arithmetic spectral flow. It fundamentally resolves the structural contradiction between the discrete natural number spectrum of number theory and the continuous modular spectrum of Type Ⅲ₁ factors in noncommutative geometry. By constructing a time-dependent evolving spectral measure, reciprocal symmetric isometric involution and global functional equation, this paper builds a continuous dynamic flow connecting discrete arithmetic systems and noncommutative modular systems. The core evolution equation of arithmetic spectral flow is derived, and the local well-posedness of the equation is verified by mollification approximation and Picard iteration. Through the research on spectral free energy functional dissipation inequality and global non-blowup estimation, this paper distinguishes finite-time collision from infinite asymptotic behavior of zeros, and proposes the global uniform spectral gap theorem. Combined with complex analytic perturbation, connected topology and Type Ⅲ₁ factor rigidity, a complete closed-loop logical argument is formed. Finally, this paper strictly proves that all non-trivial zeros of the Riemann zeta function are located on the critical line Re (s) =12. The whole research follows ZFC axiom system, functional analysis, operator algebra and complex analysis theory, without logical jumps and vague qualitative reasoning. wxsq1638@outlook. com 0009-0000-3175-1681 Xinyu Zheng

Arithmetic Spectral Flow and the Rigidity of Zeros of the Riemann Zeta Function

Key Points

Abstract

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