The global resolvent trace conjecture for the Riemann zeta function

Let $h$ be a test function with Fourier transform $\hat h$, and let $\rho$ run over the zeros of the Riemann zeta function $\zeta(s)$. Resolvent trace identity. The global resolvent trace satisfies the Weil explicit formula:

$$\sum_\rho h(\rho) = \sum_p\sum_{k\ge1}\frac{\log p}{p^{k/2}}\bigl(\hat h(k\log p)+\hat h(-k\log p)\bigr) + \text{trivial terms}.$$

This connects the proposed operator-theoretic resolvent with the explicit-formula distribution of zeta zeros; the supplied text does not establish the identity or specify its precise test-function hypotheses.