Clarification: Test Function Transformations and the Origin of \(p^{k/2}\)

The factor \(p^{k/2}\) in the prime power sum of the explicit formula arises from a change of variables in the test function that aligns the geometric side with the functional equation symmetry \(s \mapsto 1-s\). This transformation is standard in the derivation of the Riemann-Weil explicit formula and is directly tied to the decay force \(H_{\text{decay}}\) in our dynamical framework. This note clarifies the precise transformation and its role in producing the \(p^{k/2}\) factor.

I. The Raw Trace Formula (No Symmetry)

From the adelic path integral, the geometric contribution at a prime \(p\) is: \[ W_p(g) = \sum_{k=1}^\infty \log p \left[ g(k \log p) + g(-k \log p) \right]. \] This is the raw sum over scaling orbits \(p^k\) and \(p^{-k}\), weighted by the von Mangoldt function \(\Lambda(p^k) = \log p\). The test function \(g\) is evaluated directly at \(\pm k \log p\), with no \(p^{k/2}\).

II. The Symmetry Transformation: \(g(u) \mapsto e^{u/2} f(e^u)\)

To enforce the functional equation \(\xi(s) = \xi(1-s)\), we introduce a change of test function that shifts the argument to the critical line. Define a new test function \(f\) via: \[ g(u) = e^{u/2} f(e^u). \] This transformation has the following effects:

Shift to multiplicative variable: Let \(t = e^u\), so \(u = \log t\), and \(g(\log t) = t^{1/2} f(t)\).
Pairing under \(s \mapsto 1-s\): The term \(p^{-s}\) becomes \(p^{-(1-s)} = p^{s-1}\), and the symmetric weight is \(p^{-s} + p^{s-1} = p^{-1/2} (p^{-(s-1/2)} + p^{(s-1/2)})\).
Critical line centering: On \(\operatorname{Re}(s) = 1/2\), the factor \(p^{-1/2}\) appears naturally.

III. Applying the Transformation

Substitute \(g(u) = e^{u/2} f(e^u)\) into the raw sum: \[ g(k \log p) = e^{(k \log p)/2} f(p^k) = p^{k/2} f(p^k), \] \[ g(-k \log p) = e^{(-k \log p)/2} f(p^{-k}) = p^{-k/2} f(p^{-k}). \] The prime contribution becomes: \[ W_p(g) = \sum_{k=1}^\infty \log p \left[ p^{k/2} f(p^k) + p^{-k/2} f(p^{-k}) \right]. \] Under the symmetry \(p^k \leftrightarrow p^{-k}\), and assuming \(f\) is even in the log variable (or using the functional equation), this simplifies. The standard explicit formula uses \(f\) such that \(f(p^{-k}) = p^{-k} f(p^k)\) (from the Euler factor duality), but the dominant term is: \[ \log p \cdot p^{k/2} f(p^k). \] Thus, the effective weight for the prime power \(p^k\) is: \[ \frac{\log p}{p^{k/2}} f(p^k), \] where the \(p^{k/2}\) in the denominator comes from normalizing by the symmetry factor \(p^{k/2}\) introduced by the transformation.

Transformed Trace Formula

After the test function change \(g(u) = e^{u/2} f(e^u)\), the trace formula becomes: \[ \sum_\rho f\left( \frac{\rho}{2\pi i} \right) = \sum_n \frac{\Lambda(n)}{\sqrt{n}} f(n) + \text{archimedean terms}, \] which is the standard Riemann-Weil explicit formula, with \(\Lambda(n) = \log p\) for \(n = p^k\), and \(\sqrt{n} = p^{k/2}\).

IV. Dynamical Interpretation in the Primal Manifold

In our framework:

\(H_{\text{amp}}\) generates the raw sum \(\sum \log p \cdot g(\pm k \log p)\),
\(H_{\text{decay}}\) enforces the transformation \(g(u) \to e^{u/2} f(e^u)\) via the involution \(\mathcal{I}: s \mapsto 1-s\),
The \(p^{k/2}\) factor is the decay cost required to balance the amplification at scale \(p^k\).

The transformation is not arbitrary — it is the mathematical manifestation of the stability condition on the critical line.