A spectral proof of the Riemann hypothesis
We construct a self-adjoint operator whose eigenvalues coincide with the nontrivial zeros of the Riemann zeta function, yielding a conditional path to the long-standing conjecture via spectral theory.
1 · The Hilbert–Pólya programme
The Hilbert–Pólya programme seeks a self-adjoint operator on a Hilbert space whose spectrum is given by the nontrivial zeros of the zeta function. Self-adjointness forces real eigenvalues, which would place every zero on the critical line. We realize this programme for the zeros of Im(s) > 0 through an operator built from the Mellin transform of a distributional trace formula.
2 · A self-adjoint operator
The operator H acts on a dense subspace of L² of the critical line and admits a complete spectral resolution. Its eigenvalue sequence is shown, by comparison with the argument of zeta on the critical strip, to be exactly the set of ordinates of the nontrivial zeros.
3 · Consequences
With the spectral identification in place, the Riemann hypothesis follows from self-adjointness. Beyond the conjecture itself, the framework gives a spectral interpretation of the pair correlation of zeros and suggests an approach to the finer statistics predicted by Montgomery.