Topological Phase Transition in the Riemann Zeta Function: A Topological Invariant Characterising the Critical Line

Владислав Витальевич Тишков

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Topological Phase Transition in the Riemann Zeta Function: A Topological Invariant Characterising the Critical Line

В. В. Тишков

2026-07-21

https://doi.org/10.24108/preprints-3115940

We discover a topological phase transition in the Riemann zeta function. We define a topological invariant ν(α) = Ind(Kα) = (1/2πi)∮_Γ Kα'(z)/Kα(z) dz associated with the coherence kernel Kα. We prove that ν(α) = 0 for α ≠ 1/2 and ν(1/2) = 1. Thus the critical line α = 1/2 is a topological phase transition point: the topological invariant changes discontinuously from ν = 0 to ν = 1 and back. This is a new characterisation of the Riemann Hypothesis: RH is equivalent to the statement that the topological invariant ν(1/2) = 1. The phase transition is analogous to the Anderson localisation-delocalisation transition in disordered systems, the quantum Hall effect, and topological insulators. All results are rigorously proven and fully verifiable numerically.

Ссылка для цитирования:

Тишков В. В. 2026. Topological Phase Transition in the Riemann Zeta Function: A Topological Invariant Characterising the Critical Line. PREPRINTS.RU. https://doi.org/10.24108/preprints-3115940

Список литературы

1. P. W. Anderson, Absence of diffusion in certain random lattices, Phys. Rev., 109(5):1492-1505, 1958.

2. D. J. Thouless, M. Kohmoto, M. P. Nightingale, and M. den Nijs, Quantized Hall conductance in a two-dimensional periodic potential, Phys. Rev. Lett., 49(6):405-408, 1982.

3. M. Z. Hasan and C. L. Kane, Colloquium: Topological insulators, Rev. Mod. Phys., 82(4):3045-3067, 2010.

4. W. Rudin, Real and Complex Analysis, 3rd ed., McGraw-Hill, 1987.

5. G. Sierra, The Riemann zeros and the cyclic Renormalization Group, arXiv:math/0510572, 2005.

6. H. L. Montgomery, The pair correlation of zeros of the zeta function, in Analytic Number Theory, Proc. Sympos. Pure Math., Vol. 24, pp. 181-193, Amer. Math. Soc., Providence, RI, 1973.

7. M. V. Berry and J. P. Keating, The Riemann zeros and eigenvalue asymptotics, SIAM Review, 41(2):236-266, 1999.

8. A. M. Odlyzko, On the distribution of spacings between zeros of the zeta function, Math. Comp., 48:273-308, 1987.

9. E. C. Titchmarsh, The Theory of the Riemann Zeta-Function, 2nd ed., Oxford University Press, 1986.

10. H. M. Edwards, Riemann's Zeta Function, Academic Press, 1974.

11. H. Davenport, Multiplicative Number Theory, 3rd ed., Springer, 2000.

12. D. A. Goldston and H. L. Montgomery, Pair correlation of zeros and primes in short intervals, in Analytic Number Theory and Diophantine Problems, Prog. Math. 70, 183-203, Birkhäuser, 1987.

13. D. A. Goldston, S. M. Gonek, and H. L. Montgomery, Mean values of the logarithmic derivative of the Riemann zeta-function with applications to primes in short intervals, in preparation.

14. D. A. Goldston, J. Lee, J. Schettler, and A. I. Suriajaya, Pair Correlation Conjecture for the zeros of the Riemann zeta-function II: The Alternative Hypothesis, arXiv:2507.06823, 2025.

15. D. A. Goldston and A. I. Suriajaya, Zeta Zeros on the Critical Line, arXiv:2511.20059, 2025.

16. M. Cech and K. Matomaki, Pair correlation of zeros of Dirichlet L-functions: a possible path towards the conjectures of Chowla, Elliott-Halberstam and Montgomery, Math. Ann., 394, Article 43, 2026.

17. V. Tishkov, Explicit Formula for the Symmetric Coherence Kernel Associated with Dirichlet L-Functions, Preprint, 2026.

18. V. Tishkov, On the Riemann Hypothesis: A Complete Proof via the Twin Operator Method, Preprint, 2026.

19. V. Tishkov, Quantum-Ergodic Theorem for the Riemann Zeta Function, Preprint, 2026.

20. V. Tishkov, Quantum Recurrence Theorem for the Riemann Zeta Function, Preprint, 2026.

21. V. Tishkov, Temporal Correlations of the Evolution Operator for the Riemann Zeta Function, Preprint, 2026.

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