The longstanding conjecture that the nonimaginary solutions of
(1)
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where is the Riemann zeta function, are the eigenvalues of an "appropriate" Hermitian operator . Berry and Keating (1999) further conjecture that this operator is
(2)
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(3)
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where and are the position and conjugate momentum operators, respectively, and multiplication is noncommutative. Note that is symmetric but might have nontrivial deficiency indices, so while physicists define this operator to be Hermitian, mathematicians do not.