ACTA MATHEMATICA UNIVERSITATIS COMENIANAE
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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Acta Mathematica Universitatis Comenianae
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