Journal of Liberated Mathematics
Published July 6, 2025 | Version v17
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Spectral Coherence and Geometric Reformulation of the Riemann Hypothesis via Torsion-Free Vector Waves

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  • 1. Independent researcher

Description

We introduce a geometric and spectral reformulation of the Riemann Hypothesis based on the analysis of a complex vector-valued function, the Function of Residual Oscillation (FOR(N)), defined by a regularized spectral sum over the nontrivial zeros of the Riemann zeta function. This function reveals a torsion structure in the complex plane that is minimized under the critical-line condition Re(ρ) = 1/2. By analyzing the directional stability of the associated vectors, we demonstrate that the Riemann Hypothesis is equivalent to the global vanishing of the spectral torsion function τ(N). The approach combines geodesic vector dynamics, coherence cancellation, and asymptotic convergence, providing a new structural perspective on one of the most fundamental problems in mathematics.

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References

  • Titchmarsh, E.C., The Theory of the Riemann Zeta-Function, Oxford University Press, 2nd Edition, 1986.
  • Edwards, H.M., Riemann's Zeta Function, Dover Publications, 2001.
  • Bombieri, E., 'Problems of the Millennium: The Riemann Hypothesis,' Clay Mathematics Institute, 2000.
  • Ivić, A., The Riemann Zeta-Function: Theory and Applications, Dover Publications, 2003.
  • Conrey, J.B., 'The Riemann Hypothesis,' Notices of the American Mathematical Society, vol. 50, no. 3, pp. 341-353, 2003.
  • Patterson, S.J., An Introduction to the Theory of the Riemann Zeta-Function, Cambridge University Press, 1988.
  • Montgomery, H.L., 'The Pair Correlation of Zeros of the Zeta Function,' Analytic Number Theory, Proceedings of Symposia in Pure Mathematics, vol. 24, American Mathematical Society, 1973.
  • Odlyzko, A.M., 'The 1020th Zero of the Riemann Zeta Function and 175 Million of its Neighbors,' AT&T Bell Laboratories, 1989
  • Lagarias, J.C., 'Number Theory and Dynamical Systems,' Proceedings of Symposia in Applied Mathematics, vol. 46, American Mathematical Society, 1992.
  • Fujii, A., "On the distribution of the zeros of the Riemann zeta-function," Proc. Japan Acad. Ser. A Math. Sci., vol. 66, no. 7, pp. 161–166, 1990.
  • Báez-Duarte, L. (2003). A new simple proof of the Beurling-Nyman criterion for the Riemann Hypothesis. Preprint. arXiv:math/0303143
  • Korevaar, J. (2004). Tauberian Theory: A Century of Developments. Berlin: Springer.
  • Gonek, S. M. (2004). On the Linear Independence of the Ordinates of Zeros of the Riemann Zeta Function. Journal of Number Theory.
  • Weyl, H. (1916). Über die Gleichverteilung von Zahlen mod. Eins. Mathematische Annalen, 77(3), 313–352.
  • Hardy, G. H., and Littlewood, J. E. (1914). *Some problems of 'Partitio Numerorum' III: On the expression of a number as a sum of primes.* Acta Mathematica, 44(1), 1–70.
  • Hardy, G. H. (1949). *Divergent Series.* Oxford: Clarendon Press.
  • Ingham, A. E. (1932). *A Tauberian theorem for partitions.* Annals of Mathematics, 33(3), 491–498.
  • Ingham, A. E. (1932). *The Distribution of Prime Numbers.* Cambridge Tracts in Mathematics and Mathematical Physics, No. 30. Cambridge: Cambridge University Press.