This document derives the Riemann-Weil explicit formula from the trace formula developed in the previous papers. The trace formula, obtained via the adelic path integral, gives the geometric (prime) side. By equating it to the spectral trace (sum over eigenvalues of the operator \(H\)), recover the explicit formula under the Hilbert-Pólya conjecture, where the eigenvalues correspond to the imaginary parts of the zeta zeros. The derivation just reinforces the dynamical interpretation: the RH percieved as stability on the critical line.
So the operator \(H\) on \(L^2(\mathcal{P})\), decomposed as \(H = H_{\text{amp}} - H_{\text{decay}}\). Under the Hilbert-Pólya conjecture, \(H\) is self-adjoint with eigenvalues \(\gamma_n\), where the non-trivial zeros are \(\rho_n = 1/2 + i \gamma_n\) (assuming RH for real eigenvalues).
For a test function \(g: \mathbb{R} \to \mathbb{C}\) (even as rapid decay), define the operator \[ U(g) = \int_{-\infty}^\infty g(t) e^{i t H} \, dt. \] This is the Fourier transform version, motivated by quantum trace formulas (e.g., Gutzwiller). Spectrally, \[ \operatorname{Tr}(U(g)) = \sum_n \hat{g}(\gamma_n), \] where \(\hat{g}(\xi) = \int g(t) e^{i t \xi} \, dt\) is the Fourier transform (convention without 2\pi).
From the path integral trace formula (geometric side), \[ \operatorname{Tr}(U(g)) = \sum_p \sum_{k=1}^\infty \frac{\log p}{p^{k/2}} \left[ g(k \log p) + g(-k \log p) \right] + W_r(g), \] where \(W_r(g)\) is the archimedean term.
Spectral Side: Assuming the Hilbert-Pólya operator \(H\) has eigenvalues \(\gamma_n\) (heights of zeros), the trace is \[ \operatorname{Tr}(U(g)) = \sum_n \int g(t) e^{i t \gamma_n} \, dt = \sum_n \hat{g}(\gamma_n). \] To include trivial zeros and poles, extend the sum to all \(\rho\) (non-trivial zeros) plus contributions from trivial zeros (\(\rho = -2n\)) and the pole at \(s=1\).
Equating Sides: Set the spectral trace equal to the geometric trace: \[ \sum_\rho \hat{g}(\gamma) + \text{trivial/pole terms} = \sum_p \sum_k \frac{\log p}{p^{k/2}} [g(k \log p) + g(-k \log p)] + W_r(g). \] Here, \(\gamma = \operatorname{Im}(\rho - 1/2)\) (shifted by critical line).
Test Function Relation: Define \(h(u) = \hat{g}(u)\) (Fourier transform). Then the spectral side is \(\sum_\rho h(\gamma)\), where \(\gamma = \log(\operatorname{Im} \rho)\) wait, no: \(\gamma\) is the imaginary part. Standard convention: let \(h(t) = \int g(u) e^{i t u} du = \hat{g}(-t)\) (adjust sign). The prime side \(g(k \log p) = h( i k \log p )\)? Let's align. In standard explicit formula (e.g., Bombieri), for even \(h(t)\), \[ \sum_\rho h\left( \frac{\gamma}{2\pi} \right) = \int h(t) d\Pi(t) + \text{terms}, \] the common form is: \[ \sum_\rho h\left( \gamma \right) = h(0) + h(i/2) - \sum_n \frac{\Lambda(n)}{\sqrt{n}} [h(\log n) + h(-\log n)] + \int \frac{\Gamma'(1+s)}{\Gamma(1+s)} h(s) ds, \] (up to constants). Comparing, \(g(t) = h(t)\), and the prime term matches with \(\Lambda(p^k) = \log p\), \(\sqrt{p^k} = p^{k/2}\).
Functional Equation Role: The \(1/2\) shift arises from \(H_{\text{decay}}\), enforcing \(s \to 1-s\). This pairs zeros \(\rho\) and \(1-\rho\), centering on \(1/2\).
Complete Formula: include archimedean and trivial terms from \(W_r(g)\) and pole/residues, to get the Riemann-Weil explicit formula: \[ \hat{g}(0) + \hat{g}(1) - \sum_\rho \hat{g}(\rho) = \sum_p \log p \sum_{k=1}^\infty \left[ g(p^k) + p^{-k} g(p^{-k}) \right] + W_r(g), \] where adjusting \(g(u) = e^{u/2} f(e^u)\) introduces the \(p^{k/2}\) via shift (see previous clean up paper from last debate).
Trace formula is the geometric side = the spectral trace yields (explicit formula): RH is percieved as the condition for real eigenvalues.