A Curvature Decomposition of the Explicit Formula for the Riemann Zeta Function

We study the second logarithmic derivative of the completed Riemann zeta function in the form

Hξ(σ,t) = ∂²_σ log |ξ(σ + it)|²

and interpret it as a second-derivative specialization of the classical explicit formula.

This yields a prime-cutoff decomposition into local curvature contributions coming from the archimedean place and the finite Euler factors, together with a spectral remainder encoding the contribution of the nontrivial zeros.

We prove local positivity at every finite place, establish critical divergence of the truncated local curvature on the line σ = 1/2 with order (log κ)², and show convergence for σ > 1/2.

We also discuss the automorphic Rankin–Selberg analogue and the structural proximity of the curvature field to Li-type positivity criteria.

The note concludes with an open question on whether the singular kernel underlying this curvature formulation can be embedded into the admissible Weil test-function framework.