Towards a Proof of the Riemann Hypothesis: Explicit Formulas, Nyman–Beurling Approximations, and Thin-Band Integer Pairs V.SEED
Creators
Description
This repository provides the research seed package for our exploration of the Riemann Hypothesis (RH).
We reformulate RH using two equivalent perspectives:
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Explicit formulas for the Chebyshev function with truncation error control.
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Nyman–Beurling–Báez-Duarte (NB/BD) criterion, expressed as an $L^2$ approximation problem.
The key contribution is a reduction of RH to thin-band integer correlations, showing how near-diagonal pairs $(m,n)$ drive convergence in NB/BD approximations. We propose new coefficient constructions (multi-scale, phase-modulated Dirichlet polynomials) and provide numerical evidence that diagonal sign stabilization and off-diagonal suppression can be achieved simultaneously, with error terms decreasing as $N$ grows.
All results are released together with figures, CSV datasets, and LaTeX sources for reproducibility. This package is not a proof of the Riemann Hypothesis but a structural seed: a foundation for future mathematical and computational investigations.
In addition to Zenodo, the full project files are archived in GitHub under:
🔗 serabing-hash / riemann-hypothesis-project
Files
README_seed.md
Additional details
Dates
- Issued
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2025-09-29
References
- Titchmarsh, E. C. (1986). The Theory of the Riemann Zeta-function (2nd ed., revised by D. R. Heath-Brown). Oxford University Press. Edwards, H. M. (1974). Riemann's Zeta Function. Dover Publications. Báez-Duarte, L. (2003). A strengthening of the Nyman–Beurling criterion for the Riemann hypothesis. Rendiconti del Circolo Matematico di Palermo, 52(3), 375–380. https://doi.org/10.1007/s12215-003-0007-1 Conrey, J. B. (2003). The Riemann Hypothesis. Notices of the American Mathematical Society, 50(3), 341–353. Ivić, A. (1985). The Riemann Zeta-Function: Theory and Applications. Dover Publications.