Relative Trace Formula: Bridging Automorphic Spectra and Analytic Number Theory — E8 Intelligence Research

Abstract

FINDING: The relative trace formula (RTF) bridges automorphic spectrum asymptotics and analytic number theory, with Kuznetsov-type variants yielding sharp spectral averages over arithmetic surfaces. | MATH: Trace formula identity: \(\sum_{\pi} \mathrm{tr}(\pi(f)) \approx \sum_{\gamma} \widehat{f}(\gamma)\) (geometric side) vs. spectral side \(\sum_{\pi} \mathrm{vol}(\Gamma_\pi) \cdot \mathcal{P}(\pi)\); Kuznetsov-type RTF: \(\sum_{j} \frac{\alpha_j}{\cosh(\pi t_j)} |\mathcal{P}(\pi_j)|^2 = \text{geometric terms} + \text{continuous spectrum}\). Key constants: spectral parameters \(t_j\) (eigenvalues \(\lambda_j = \frac{1}{4} + t_j^2\) for \(SL_2(\mathbb{Z})\backslash\mathbb{H}\)); asymptotic Weyl law \(N(T) \sim \frac{\mathrm{vol}(\Gamma\backslash\mathbb{H})}{4\pi} T^2\). | CONNECTION: The spectral parameters \(t_j\) are real for Maass forms — the Hilbert-Polya operator would have eigenvalues \(t_j\) on the critical line \(\Re(s)=1/2\). The trace formula's geometric side contains length Author: Andrew Stewart Caldin, Independent Researcher, UK. Part of the E8 Intelligence Research series. Platform: e8intelligence.com