The factor \(p^{k/2}\) in the prime power terms of the trace formula is subtle: It is not an ad-hoc idea but emerges from the symmetry under \(s \mapsto 1-s\) — the functional equation of the zeta function is enforced by the decay force \(H_{\text{decay}}\) and the critical line stability condition. This resolves the debate in the earlier proof sketch, grounding the \(p^{k/2}\) factor in the dynamical balance of amplification and decay.
Consider the local contribution at a prime \(p\) in the adelic trace integral, before imposing the functional equation symmetry. The scaling orbits are labeled by \(p^k\) (\(k \in \mathbb{Z}\)), and the raw sum over fixed points gives: \[ W_p^{\text{raw}}(g) = \sum_{k \neq 0} \log p \cdot g(\log |p^k|) = \sum_{k=1}^\infty \log p \left[ g(\log p^k) + g(\log p^{-k}) \right]. \] This is the unbalanced geometric side, reflecting only the multiplicative structure (Euler product) via \(H_{\text{amp}}\). There is no \(p^{k/2}\) here — only \(\log p\) weights and the test function evaluated at \(\pm \log p^k\).
The decay force \(H_{\text{decay}}\) is derived from the involution \(\mathcal{I}: s \mapsto 1-s\), which maps the idele norm \(|j| \mapsto |j|^{-1}\). This symmetry acts on the full space \(C_{\mathbb{Q}} = \mathbb{R}_{>0} \times \mathcal{P}\) and induces a pairing between orbits: \[ \text{Orbit at } p^k \quad \leftrightarrow \quad \text{Orbit at } p^{-k}. \] The kernel \(K_{\text{decay}}(x,y)\) couples these orbits with a weight that reflects the change in scale under \(\mathcal{I}\). Specifically, the decay operator introduces a factor of \(|j|^{-1/2}\) to balance the amplification at \(|j|\).
In the trace formula, this symmetry is enforced by averaging over the paired orbits with the critical line shift. The stable eigenstates (zeros) live on \(\operatorname{Re}(s) = 1/2\), so the test function \(g\) is evaluated at the midpoint of the paired logs: \[ \log p^k \quad \text{and} \quad \log p^{-k} = -\log p^k \quad \Rightarrow \quad \text{midpoint} = 0, \] but more importantly, the weight of the contribution is shifted by the symmetry factor.
To make this precise, we use the completed zeta function and the functional equation. Define the test function in terms of the complex variable \(s\): \[ \hat{g}(s) = \int_{-\infty}^\infty g(\lambda) e^{i \lambda s} \, d\lambda. \] The spectral side of the explicit formula is: \[ \sum_\rho \hat{g}(\rho) + \sum_\rho \hat{g}(1-\rho), \] where \(\rho\) are the non-trivial zeros. Under the functional equation \(\xi(s) = \xi(1-s)\), the zeros come in pairs \(\rho\) and \(1-\rho\), and the symmetry forces: \[ \hat{g}(\rho) + \hat{g}(1-\rho) = 2 \operatorname{Re} \hat{g}(\rho). \] On the critical line, \(\rho = 1/2 + i\gamma\), so \(1-\rho = 1/2 - i\gamma\), and the pair sums to a real contribution.
On the geometric side, the prime power \(p^k\) corresponds to a term in the Dirichlet series at \(n = p^k\). The Euler factor is: \[ (1 - p^{-s})^{-1} = \sum_{k=0}^\infty p^{-k s}. \] Under \(s \mapsto 1-s\), this becomes: \[ (1 - p^{s-1})^{-1} = \sum_{k=0}^\infty p^{-k(s-1)} = \sum_{k=0}^\infty p^{k(1-s)}. \] The symmetric combination that respects the functional equation is the average: \[ \frac{1}{2} \left[ (1 - p^{-s})^{-1} + (1 - p^{s-1})^{-1} \right]. \] For the non-trivial part (\(k \geq 1\)), the cross terms yield: \[ p^{-k s} + p^{-k(1-s)} = p^{-k s} + p^{-k + k s} = p^{-k/2} (p^{-k(s - 1/2)} + p^{k(s - 1/2)}). \] Thus, the symmetric weight for the prime power \(p^k\) is: \[ p^{-k/2} \cdot 2 \cosh(k (s - 1/2) \log p). \] On the critical line \(s = 1/2\), this simplifies to: \[ p^{-k/2} \cdot 2. \] Therefore, the contribution per prime power in the explicit formula is: \[ \frac{\log p}{p^{k/2}} \left[ g(k \log p) + g(-k \log p) \right], \] where the \(p^{-k/2}\) comes from the functional equation symmetry, and the factor of 2 is absorbed into the test function normalization.
The geometric side of the trace formula is: \[ \operatorname{Tr}(U(g)) = \sum_p \sum_{k=1}^\infty \frac{\log p}{p^{k/2}} \left[ g(k \log p) + g(-k \log p) \right] + W_r(g), \] where:
In the Primal Manifold framework: