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2011

The Berry-Keating Hamiltonian and the Local Riemann Hypothesis

Srednicki, Mark

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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.

The Berry-Keating Hamiltonian and the Local Riemann Hypothesis · Around

Built on

Tate J 1950 Fourier analysis in number fields and Hecke’s zeta functions, Princeton PhD thesis, reprinted in 1967 Algebraic number theory
1967
Earlier work this paper cites.
Edwards H M 1974 Riemann’s Zeta Function
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Earlier work this paper cites.
Koblitz N 1984 p-adic Numbers, p-adic Analysis, and Zeta-Functions
1984
Earlier work this paper cites.
Bump D and Ng E K-S 1986 On Riemann’s zeta function Math. Z
1986
Earlier work this paper cites.

Similar

Vladimirov V S, Volovich I V and Zelenov E I 1994 p-adic Analysis and Mathematical Physics
1994
Cited alongside, same era.
Berry M V and Keating J P 1999 H = x p H=xp and the Riemann zeros, in Supersymmetry and Trace Formulae: Chaos and Disorder
1999
Cited alongside, same era.
Bump D, Choi K-K, Kurlberg P and Vaaler J 2000 A local Riemann hypothesis I Math. Z
2000
Cited alongside, same era.

Then

Kurlberg P 2000 A local Riemann hypothesis II Math. Z
2000
Later among the works it cites.
Derbyshire J 2004 Prime Obsession
2004
Later among the works it cites.
Sierra G 2005 The Riemann zeros and the cyclic renormalization group J. Stat. Mech
Original
2010
Later among the works it cites.

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