A Strategy for Proving Riemann Hypothesis

A strategy for proving Riemann hypothesis is suggested. The vanishing of the Rieman Zeta reduces to an orthogonality condition for the eigenfunctions of a non-Hermitian operator D+ having the zeros of Riemann Zeta as its eigenvalues. The construction of D+ is inspired by the conviction that Riemann Zeta is associated with a physical system allowing conformal transformations as its symmetries. The eigenfunctions of D+ are analogous to the so called coherent states and in general not orthogonal to each other. The states orthogonal to a vacuum state (which has a negative norm squared) correspond to the zeros of the Riemann Zeta. The induced metric in the space V of states which correspond to the zeros of the Riemann Zeta at the critical line Re[s] = 1/2 is hermitian and hermiticity requirement actually implies Riemann hypothesis. Conformal invariance in the sense of gauge invariance allows only the states belonging to V. Riemann hypothesis follows also from a restricted form of a dynamical conformal invariance in V. ⋆ I want to express my deep gratitude to Dr. Matthew Watkins for providing me with information about Riemann Zeta and for generous help, in particular for reading the earlier versions of the work and pointing out several inaccuracies and errors. I am also grateful for Prof. Masud Chaichian and Doc. Claus Montonen for encouraging comments and help. Present address: Kadermonkatu 16, 10900, Hanko, Finland. Email: [email protected]