Vol. LXXII, 1(2003)
p. 1 13

A Strategy for Proving Riemann Hypothesis
M. Pitkanen

Abstract.  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 $\Rea[s]=1/2$ 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 $V$.

AMS subject classification:  00-XX, Secondary 81-XX
Keywords:  Riemann hypothesis, conformal invariance

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