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Journal of Operator Theory

Volume 85, Issue 1, Winter 2021  pp. 259-278.

The scaling Hamiltonian

Authors:  Alain Connes (1), Caterina Consani (2)
Author institution: (1) Coll\'ege de France, 3 rue Ulm, 75005, Paris, France, I.H.E.S. 91440 Bures-sur-Yvette, France, \emph{and} Department of Mathematics, Ohio State University, Columbus, OH 43210, U.S.A.
(2) Department of Mathematics, The Johns Hopkins University, Baltimore MD, 21218, U.S.A.

Summary: We reconcile, at the semi-classical level, the original spectral realization of zeros of the Riemann zeta function as an ``absorption'' picture using the ad\`ele class space, with the ``emission'' semi-classical computations of Berry and Keating. We then use the quantized calculus to analyse the recent attempt of X.-J.~Li at proving Weil's positivity, and explain its limit. Finally, we propose an operator theoretic semi-local framework directly related to the Riemann hypothesis.

Keywords: Riemann zeta function, Hamiltonian, semi-local trace formula, Weil positivity

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