📝 SummarXiv

Abstract

Define the completed eta function $ Z(s) := \Gamma(s+1)\, (1-2^{1-s}) \, \zeta(s) $, and denote by $ Z_{set} := \left\{\gamma \;\middle|\; Z(\gamma)=0 \right\} $ its set of zeros, which includes both the periodic Dirichlet zeros and the nontrivial Riemann zeros $\rho$. We introduce an unbounded \emph{non-symmetric} operator \[ R \colon D(R) \subset L^2([0,\infty)) \to L^2([0,\infty)) \, , \] with spectrum \[ \sigma(R) = \left\{ i\left(1/2- \gamma \right) \;\middle|\; \gamma \in Z_{set} \right\} \, . \] We first show that the existence of a rank-one Riesz projector entails $ \zeta'(\gamma)\neq 0 $ for all $ \gamma $, and in particular for the nontrivial zeros $\rho$, thereby establishing their simplicity. Next, we construct a positive semidefinite operator $ W $ satisfying \[ R^\dagger W = W R \, . \] The eigenstates corresponding to the periodic Dirichlet zeros lie in the kernel of $W$, while its positive semidefiniteness enforces $ \Re(\rho)=1/2$, in accordance with the Riemann Hypothesis. Finally, based on $W$, we define a self-adjoint operator whose spectrum consists precisely of the imaginary parts of the nontrivial Riemann zeros.