A Mesoscopic Repulsion Bound for Zeros of the Riemann Zeta Function via the Explicit Formula
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| Sprache: | Englisch |
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2026
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| _version_ | 1866901186611773440 |
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| author | Farman, Mohhomad |
| author_facet | Farman, Mohhomad |
| contents | <p>I establish a lower bound on the localized pair correlation of non-trivial zeros of the Riemann zeta function ζ(s) in a mesoscopic window [T, T+H] with H ≥ T^θ for a fixed θ > 0. Using only the classical explicit formula and the positivity of a Fourier-weighted sum over primes, we prove that the pair correlation measure R₂(u; T, H) satisfies a quadratic vanishing lower bound near the origin — a signature of zero repulsion. Specifically, we show that ∫_{|u| ≤ U} R₂(u; T, H) du ≥ c·U³ for some absolute constant c > 0 and all sufficiently small U > 0.<br><br>The argument requires no unproven hypotheses: neither the Riemann Hypothesis nor the GUE conjecture is assumed. A corollary provides a weak average spacing bound of order N(T,H)^{−1/3}. We discuss the relationship with random matrix theory predictions and directions for future work.<br><br>MSC 2020: 11M26, 11M06</p> |
| format | Recurso digital |
| id | zenodo_https___doi_org_10_5281_zenodo_19373941 |
| institution | Zenodo |
| language | eng |
| publishDate | 2026 |
| publisher | Zenodo |
| record_format | zenodo |
| spellingShingle | A Mesoscopic Repulsion Bound for Zeros of the Riemann Zeta Function via the Explicit Formula Farman, Mohhomad Riemann zeta function pair correlation zero repulsion explicit formula mesoscopic statistics analytic number theory GUE conjecture Montgomery conjecture L-functions random matrix theory <p>I establish a lower bound on the localized pair correlation of non-trivial zeros of the Riemann zeta function ζ(s) in a mesoscopic window [T, T+H] with H ≥ T^θ for a fixed θ > 0. Using only the classical explicit formula and the positivity of a Fourier-weighted sum over primes, we prove that the pair correlation measure R₂(u; T, H) satisfies a quadratic vanishing lower bound near the origin — a signature of zero repulsion. Specifically, we show that ∫_{|u| ≤ U} R₂(u; T, H) du ≥ c·U³ for some absolute constant c > 0 and all sufficiently small U > 0.<br><br>The argument requires no unproven hypotheses: neither the Riemann Hypothesis nor the GUE conjecture is assumed. A corollary provides a weak average spacing bound of order N(T,H)^{−1/3}. We discuss the relationship with random matrix theory predictions and directions for future work.<br><br>MSC 2020: 11M26, 11M06</p> |
| title | A Mesoscopic Repulsion Bound for Zeros of the Riemann Zeta Function via the Explicit Formula |
| topic | Riemann zeta function pair correlation zero repulsion explicit formula mesoscopic statistics analytic number theory GUE conjecture Montgomery conjecture L-functions random matrix theory |
| url | https://doi.org/10.5281/zenodo.19373941 |