Unconditional Density Bounds for Quadratic Norm-Form Energies via Lorentzian Spectral Weights

Abstract

For a real quadratic field $\mathbb{Q}(\sqrt{d})$, we study the norm-form energy $\mathcal{N} = S_ζ^2 - d \cdot S_L^2$, where $S_ζ$ and $S_L$ are Lorentzian-weighted zero sums with $w(ρ) = 2/(1/4 + γ^2)$. We prove three main results. (1) Spacelike spectral data: $\mathcal{N} < 0$ 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 $M$, $\mathrm{dens}\{\mathcal{N} > 0\} \leq 2\|f_{S_L^{(M)}}\|_\infty \cdot (W_1(ζ)/\sqrt{d} + \varepsilon_M)$, established unconditionally via Jacobi--Anger resonance analysis. (3) Exact asymptotic: under the computationally verified hypothesis that the infinite resonance lattice $Λ_\infty$ has finite rank (verified for $M \leq 20$, where rank $= 0$), the sharp asymptotic $\mathrm{dens}\{\mathcal{N} > 0\} = C(d)/\sqrt{d} + o(1/\sqrt{d})$ holds. For $d = 5$, $C(5) = 2 f_{S_L}(0) \cdot \mathbb{E}[|S_ζ|] = 0.1191$; 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 the first 1000 zeros at 70 decimal places for $L(s,χ_d)$ with $d \in \{2,3,5,6,7,10,11,13\}$, all rigorously certified by ARB interval arithmetic.

Figures (2)

• Figure 1: The norm form $N(\delta) = S_\zeta(\delta)^2 - 5\,S_L(\delta)^2$ for $d = 5$, computed using the first 100 zeros of each function with Lorentzian weights $w(\gamma) = 2/(\frac{1}{4} + \gamma^2)$. Panel (a): the oscillating signal over $[0,200]$, showing that $N$ is overwhelmingly negative. Panel (b): the local fraction of $\delta$-values where $N > 0$, computed in sliding windows across $[0,2000]$; the measure of the set where $N > 0$ is approximately $5.5\%$ over $[0,2000]$, consistent with the predicted $O(1/\sqrt{d})$ density.
• Figure 2: Convergence of the spectral density $f_{S_L^{(M)}}(0)$ as a function of truncation level $M$ for $d = 5$ (conductor $5$). Blue circles: the density at the origin, computed as $(1/\pi)\int_0^\infty \prod_{k=1}^M J_0(b_k t)\, dt$. Gray squares: the sup-norm bound $\|f_{S_L^{(M)}}\|_\infty \leq (1/\pi)\int_0^\infty \prod_{k=1}^M |J_0(b_k t)|\, dt$. Dashed red line: the ARB-certified limiting value $f_{S_L^{(20)}}(0) = 8.3129$. The density stabilizes rapidly, with corrections from zeros beyond $M = 7$ contributing less than $1\%$.

Theorems & Definitions (140)

• Remark 1.1: Purpose of Appendix F
• Definition 2.1: Lorentzian weight
• Proposition 2.2: Critical line maximality
• proof
• Definition 2.3: Weighted zero sums
• Proposition 2.4: Explicit value of $S_\zeta$
• Theorem 2.5: Trudgian Trudgian2012
• proof : Proof of Proposition \ref{['prop:S-zeta-value']}
• Definition 2.6: Norm-form energy
• Definition 2.7: Causal classification