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arXiv · 1104.1850

The Berry-Keating Hamiltonian and the Local Riemann Hypothesis

Mark Srednicki
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Abstract

The local Riemann hypothesis states that the zeros of the Mellin transform of a harmonic-oscillator eigenfunction (on a real or p-adic configuration space) have real part 1/2. For the real case, we show that the imaginary parts of these zeros are the eigenvalues of the Berry-Keating hamiltonian H=(xp+px)/2 projected onto the subspace of oscillator eigenfunctions of lower level. This gives a spectral proof of the local Riemann hypothesis for the reals, in the spirit of the Hilbert-Polya conjecture. The p-adic case is also discussed.

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Mark Srednicki. 2011-04-11. The Berry-Keating Hamiltonian and the Local Riemann Hypothesis. https://doi.org/10.1088/1751-8113/44/30/305202

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