Nontrivial Riemann Zeros as Spectrum

Define $\Upsilon(s) := \Gamma(s+1)\, (1-2^{1-s}) \, \zeta(s)$ , and denote by $\mathcal{Z} := \left\{\gamma \in \mathbb{C} \;\middle|\; \Upsilon (\gamma)=0 \right\}$ its set of zeros, which includes both the periodic eta zeros, determined by $(1-2^{1-s}) =0$ with $s \neq 1$ , and the nontrivial zeta zeros $\rho$ . We introduce a non-symmetric operator

$\displaystyle \hat{\mathcal{R}} \colon \mathcal{D}(\hat{\mathcal{R}}) \subset L^2([0,\infty)) \to L^2([0,\infty)) \, ,$

with spectrum

$\displaystyle \sigma(\hat{\mathcal{R}}) = \left\{ i\left(1/2- \gamma \right) \;\middle|\; \gamma \in \mathcal{Z} \right\} \, .$

Assuming that all nontrivial zeros of the Riemann zeta function are simple, we construct a positive semidefinite operator $\hat{W}$ intertwining $\hat{\mathcal{R}}$ and its adjoint on the spectral subspace associated with the nontrivial zeros, $\hat{\mathcal{R}}^\dagger \hat{W} = \hat{W} \hat{\mathcal{R}}$ . The positivity of $\hat{W}$ , which represents an operator-theoretic form of (Bombieri’s refinement of) Weil’s positivity criterion, enforces $\Re(\rho)=1/2$ for all $\rho$ , in accordance with the Riemann Hypothesis. Furthermore, from the similarity between $\hat{\mathcal{R}}$ and $\hat{\mathcal{R}}^\dagger$ , we obtain a self-adjoint operator, whose spectrum coincides with the imaginary parts of the nontrivial zeta zeros.

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Nontrivial Riemann Zeros as Spectrum