Hilbert and Polya suggested a physical way to verify the Riemann hypotesis about $\zeta(x)$. If the Riemann hypotesis is true, we can state all eigenvalues of physical problems are real. What is the connection between the eigenvalues and the $\zeta$ function?
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2$\begingroup$ Related: physics.stackexchange.com/q/26856/2451 and links therein. $\endgroup$ – Qmechanic♦ Jul 3 '14 at 12:04
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In http://arxiv.org/abs/1608.03679 , the authors consider a Hamiltonian
$\hat{H}=\frac{1}{I-\exp(-i\hat{p})}(\hat{x}\hat{p}+\hat{p}\hat{x})(I-\exp(-i\hat{p}))$,
where $I$ is the identity matrix (I suppose - I cannot type the symbol they use), and claim that "a formal calculation of the eigenstates $\{\psi_n\}$ and eigenvalues $\{E_n\}$ of $\hat{H}$ shows that with the boundary condition $\psi_n(0)=0$ for all $n$ the eigenvalues satisfy the property that $\{\frac{1}{2}(1-iE_n)\}$ are the nontrivial zeros of the Riemann zeta function. This non-Hermitian form of the Hilbert-Polya conjecture thus confirms the Berry- Keating conjecture."
