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2025
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| Online Access: | https://doi.org/10.5281/zenodo.17622278 |
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| author | liu, gongshan |
| author_facet | liu, gongshan |
| contents | <p>Lehmer pairs—consecutive Riemann zeros with unusually small spacing—have<br>been studied since 1956, yet their internal structure remains poorly understood.<br>We introduce a geometric invariant I = �<br>p∈{5,7,11} log(p) · (ϕp − 1<br>2)2, where ϕp are<br>phase parameters from the explicit formula. This simple quadratic form predicts<br>21.5% of spacing variance within Lehmer pairs (ρ = 0.474, p < 10−20, n = 336),<br>revealing quantifiable continuous structure.<br>Most remarkably, spacing obeys a thermodynamic-like state equation:<br>∆γ = (0.338 ± 0.021) + (0.642 ± 0.072) · I<br>(1)<br>with R2 = 0.215 (bootstrap validated). The geometric invariant I exhibits scale<br>invariance across zero heights γ ∈ [79, 9875] (CV=0.107), maintaining predictive<br>power where direct use of |ζ′(1/2+iγ)| degrades. Pairs with tighter spacing cluster<br>near optimal phase configuration (0.5, 0.5, 0.5) with 1.66× lower deviation (Cohen’s<br>d = 0.92, p < 10−14).<br>The state equation connects microscopic (prime oscillations) to macroscopic<br>(zero spacing) scales through an order parameter, providing the first quantitative<br>characterization of continuous variation within the discrete Lehmer phenomenon.<br>Results remain robust under cross-validation (CV R2 = 0.201), permutation testing<br>(p < 10−4, Z = 9.1σ), and alternative platform definitions.</p> |
| format | Recurso digital |
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| institution | Zenodo |
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| publishDate | 2025 |
| publisher | Zenodo |
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| spellingShingle | A Thermodynamic State Equation for Riemann Zero Spacing: Phase-Space Structure of Lehmer Pairs liu, gongshan <p>Lehmer pairs—consecutive Riemann zeros with unusually small spacing—have<br>been studied since 1956, yet their internal structure remains poorly understood.<br>We introduce a geometric invariant I = �<br>p∈{5,7,11} log(p) · (ϕp − 1<br>2)2, where ϕp are<br>phase parameters from the explicit formula. This simple quadratic form predicts<br>21.5% of spacing variance within Lehmer pairs (ρ = 0.474, p < 10−20, n = 336),<br>revealing quantifiable continuous structure.<br>Most remarkably, spacing obeys a thermodynamic-like state equation:<br>∆γ = (0.338 ± 0.021) + (0.642 ± 0.072) · I<br>(1)<br>with R2 = 0.215 (bootstrap validated). The geometric invariant I exhibits scale<br>invariance across zero heights γ ∈ [79, 9875] (CV=0.107), maintaining predictive<br>power where direct use of |ζ′(1/2+iγ)| degrades. Pairs with tighter spacing cluster<br>near optimal phase configuration (0.5, 0.5, 0.5) with 1.66× lower deviation (Cohen’s<br>d = 0.92, p < 10−14).<br>The state equation connects microscopic (prime oscillations) to macroscopic<br>(zero spacing) scales through an order parameter, providing the first quantitative<br>characterization of continuous variation within the discrete Lehmer phenomenon.<br>Results remain robust under cross-validation (CV R2 = 0.201), permutation testing<br>(p < 10−4, Z = 9.1σ), and alternative platform definitions.</p> |
| title | A Thermodynamic State Equation for Riemann Zero Spacing: Phase-Space Structure of Lehmer Pairs |
| url | https://doi.org/10.5281/zenodo.17622278 |