Quantum Prime Spectral Theory (QPST)
Description
This work presents a operator-theoretic formulation of the Riemann explicit formula based on a canonical self-adjoint operator combining arithmetic and archimedean components.
The arithmetic part has discrete spectrum given by logarithms of integers, while the archimedean part is defined in terms of the generator of dilations and carries continuous spectrum. Using regularized resolvent traces, we show that the explicit formula arises naturally in the classical Titchmarsh–Mellin form.
In this framework, the von Mangoldt weights and the Gamma factor originate from distinct spectral mechanisms, and the non-trivial zeros of the Riemann zeta function appear as spectral singularities of the resolvent rather than as genuine eigenvalues. The construction is fully canonical and clarifies the structural role of continuous spectrum and spectral regularization, without making any new claim concerning the Riemann Hypothesis.
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Additional details
Additional titles
- Subtitle (English)
- A Canonical Spectral Framework for the Hilbert–Pólya Paradigm
Dates
- Updated
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2026-02-06Updated to Version VII
References
- E. C. Titchmarsh, The Theory of the Riemann Zeta-Function, 2nd ed., revised by D. R. Heath-Brown, Oxford University Press, 1986.
- H. M. Edwards, Riemann's Zeta Function, Dover, 2001.
- H. Iwaniec and E. Kowalski, Analytic Number Theory, American Mathematical So- ciety Colloquium Publications, Vol. 53, 2004.
- A. Weil, "Sur les formules explicites de la théorie des nombres premiers," Comm. Sém. Math. Univ. Lund (1952), 252–265.
- A. P. Guinand, "A summation formula in the theory of prime numbers," Proc. London Math. Soc. (2) 50 (1948), 107–119.
- A. Connes, "An essay on the Riemann hypothesis," arXiv:math/0005262 (2000).
- M. Reed and B. Simon, Methods of Modern Mathematical Physics I: Functional Anal- ysis, Academic Press, 1980.
- M. Reed and B. Simon, Methods of Modern Mathematical Physics II: Fourier Anal- ysis, Self-Adjointness, Academic Press, 1975.
- N. Dunford and J. T. Schwartz, Linear Operators, Part II: Spectral Theory, Inter- science, 1963.
- M. H. Stone, "On one-parameter unitary groups in Hilbert space," Ann. of Math. 33 (1932), 643–648.
- G. Bennett, "Schur multipliers," Duke Math. J. 44 (1977), 603–639.
- M. V. Berry and J. P. Keating, "The Riemann zeros and eigenvalue asymptotics," SIAM Rev. 41 (1999), 236–266.
- A. M. Odlyzko, "On the distribution of spacings between zeros of the zeta function," Math. Comp. 48 (1987), 273–308.