Expository notes on Spectral Reciprocity with Explicit Transform

Abstract

We assemble three basic analytic inputs -- the Kuznetsov trace formula on $\mathrm{SL}_2(\mathbb Z)$ with explicit continuous spectrum, the $\mathrm{GL}_3$ Voronoi formula, and $t$-aspect second-moment bounds for $L(1/2+it,\varphi)$ -- into a single framework for a smoothed $\mathrm{GL}_3$ spectral average. For a fixed Hecke-Maass cusp form $\varphi$ on $\mathrm{SL}_3(\mathbb Z)$, we evaluate a weight-$0$ spectral average of $L(1/2,\varphi\times f_j)$ over the $\mathrm{GL}_2$ Maass spectrum. In the Kuznetsov normalization where the diagonal transform has density $t\tanh(\pi t)$, the diagonal contributes exactly $2\mathcal H_0[h]$; the off-diagonal and the continuous spectrum are bounded with power savings consistent with the currently best unconditional second-moment bounds in the $\mathrm{GL}_3$ $t$-aspect. The argument is organized into a sequence of steps: normalizations and approximate functional equations, insertion into Kuznetsov, Voronoi summation on $\mathrm{GL}_3$, a bilinear estimate for the off-diagonal, evaluation of the diagonal and the Eisenstein contribution, moment refinements and parameter optimization, and finally plateau-smooth spectral windows and standard generalizations.