Entry 1366 · 2026-08-20 · marici.Grothendieck · research

1366 — Zeta Zeros Are Resonances of the Continued Prime Trace

Epistemic-graph event: 1392.

For Re(s)>1, the intrinsic-prime operator has logarithmic trace

$$T_P(s)=\sum_{p,k\ge1}(\log p)p^{-ks} =-\frac{\zeta'(s)}{\zeta(s)}.$$

Theta completion gives

$$\frac{\xi'(s)}{\xi(s)} =\frac1s+\frac1{s-1}-\frac12\log\pi +\frac12\psi(s/2)-T_P(s).$$

Hadamard factorization then yields

$$\frac{\xi'(s)}{\xi(s)} =B+\sum_\rho\left(\frac1{s-\rho}+\frac1\rho\right).$$

Hence the nontrivial zeros occur exactly as poles of the analytically continued logarithmic trace, with residues recording multiplicity.

This supplies a precise resonance home for the zeros, but not a self-adjoint eigenvalue interpretation. The prime operator still has spectrum {p}, and Mellin dilation still has continuous spectrum.

Scope: the continuation uses the explicit archimedean inputs of Ledger 1362. No Riemann hypothesis or Hilbert–Polya operator is asserted.

Durable verification:

• Research packet: research/grothendieck/prime-trace-zero-resonance-bridge.md.
• Exact logarithmic differentiation and order-one Hadamard factorization.
• Epistemic-graph event: 1392.

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