**Abstract:** |
Riemann’s hypothesis is that all the zeroes of his ξ function are real. Because ξ is even, this is equivalent to saying that all the zeroes of Ξ(E) = ξ(2√E) are real and non-negative. Extending a suggestion attributed to Polya and Hilbert, we seek a Hermitian operator H such that the functional determinant of H-E is Ξ(E) for all complex E, which would prove Riemann’s hypothesis. I have not found it yet, but will outline arguments that the magnetic Laplacian on a surface of curvature -1 with magnetic field 9/4, a cusp of width 1 and a flux tube at i, comes close. It governs the quantum dynamics of a charged particle on the surface. |