Unconditional Density Bounds for Quadratic Norm-Form Energies via Lorentzian Spectral Weights
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| Formato: | Preprint |
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2026
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| _version_ | 1866915856109273088 |
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| author | Shiller, Peter |
| author_facet | Shiller, Peter |
| contents | For a real quadratic field $\mathbb{Q}(\sqrt{d})$, we study the norm-form energy $N = S_ζ^2 - d \cdot S_L^2$, where $S_ζ$ and $S_L$ are Lorentzian-weighted zero sums with $w(ρ) = 2/(\tfrac{1}{4} + γ^2)$. We prove three main results.
(1) Spacelike spectral data: $N < 0$ unconditionally for all squarefree $d > 1$, as a consequence of a low-lying zero dominance theorem proved via explicit zero-counting.
(2) Effective density bound: at each verified truncation level $M$, $\mathrm{dens}\{N > 0\} \leq 2\|f_{S_L^{(M)}}\|_\infty \cdot (W_1(ζ)/\sqrt{d} + ε_M)$, established unconditionally via Jacobi--Anger resonance analysis. At fixed $M$ the bound is nontrivial only for sufficiently large $d$; the $O(1/\sqrt{d})$ rate requires $M$ to grow with $d$, which in turn requires a uniform density bound that we establish under a computationally verified finite-rank condition on the resonance lattice.
(3) Exact asymptotic: under the computationally verified hypothesis that the infinite resonance lattice $Λ_\infty$ has finite rank (verified to have rank $0$ for $M \leq 20$), the sharp asymptotic $\mathrm{dens}\{N > 0\} = C(d)/\sqrt{d} + o(1/\sqrt{d})$ holds. For $d = 5$, $C(5) = 2\,f_{S_L}(0)\cdot\mathbb{E}[|S_ζ|] = 0.1193$; the constant depends on $d$ through the zeros of $L(s,χ_d)$, and $C(d) = O(1/\log d)$ as $d \to \infty$.
Appendix F tabulates between 1004 and 1044 zeros at 70 decimal places for $L(s,χ_2)$, $L(s,χ_3)$, $L(s,χ_5)$, $L(s,χ_6)$, $L(s,χ_7)$, $L(s,χ_{10})$, $L(s,χ_{11})$, and $L(s,χ_{13})$, all rigorously certified by ARB interval arithmetic. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2603_00301 |
| institution | arXiv |
| publishDate | 2026 |
| record_format | arxiv |
| spellingShingle | Unconditional Density Bounds for Quadratic Norm-Form Energies via Lorentzian Spectral Weights Shiller, Peter Number Theory 11M06 (Primary) 11M26, 11R42, 42A75 (Secondary) For a real quadratic field $\mathbb{Q}(\sqrt{d})$, we study the norm-form energy $N = S_ζ^2 - d \cdot S_L^2$, where $S_ζ$ and $S_L$ are Lorentzian-weighted zero sums with $w(ρ) = 2/(\tfrac{1}{4} + γ^2)$. We prove three main results. (1) Spacelike spectral data: $N < 0$ unconditionally for all squarefree $d > 1$, as a consequence of a low-lying zero dominance theorem proved via explicit zero-counting. (2) Effective density bound: at each verified truncation level $M$, $\mathrm{dens}\{N > 0\} \leq 2\|f_{S_L^{(M)}}\|_\infty \cdot (W_1(ζ)/\sqrt{d} + ε_M)$, established unconditionally via Jacobi--Anger resonance analysis. At fixed $M$ the bound is nontrivial only for sufficiently large $d$; the $O(1/\sqrt{d})$ rate requires $M$ to grow with $d$, which in turn requires a uniform density bound that we establish under a computationally verified finite-rank condition on the resonance lattice. (3) Exact asymptotic: under the computationally verified hypothesis that the infinite resonance lattice $Λ_\infty$ has finite rank (verified to have rank $0$ for $M \leq 20$), the sharp asymptotic $\mathrm{dens}\{N > 0\} = C(d)/\sqrt{d} + o(1/\sqrt{d})$ holds. For $d = 5$, $C(5) = 2\,f_{S_L}(0)\cdot\mathbb{E}[|S_ζ|] = 0.1193$; the constant depends on $d$ through the zeros of $L(s,χ_d)$, and $C(d) = O(1/\log d)$ as $d \to \infty$. Appendix F tabulates between 1004 and 1044 zeros at 70 decimal places for $L(s,χ_2)$, $L(s,χ_3)$, $L(s,χ_5)$, $L(s,χ_6)$, $L(s,χ_7)$, $L(s,χ_{10})$, $L(s,χ_{11})$, and $L(s,χ_{13})$, all rigorously certified by ARB interval arithmetic. |
| title | Unconditional Density Bounds for Quadratic Norm-Form Energies via Lorentzian Spectral Weights |
| topic | Number Theory 11M06 (Primary) 11M26, 11R42, 42A75 (Secondary) |
| url | https://arxiv.org/abs/2603.00301 |