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What is the analytic continuation of the Riemann Zeta ..
The Riemann zeta function plays a crucial role in number theory as well as physics. Indeed, the distribution of primes is intimately connected to the non-trivial zeros of this function. We briefly summarize the essential properties of the Riemann zeta function and then present a quantum mechanical system which when measured appropriately yields zeta. We emphasize that for the representation in terms of a Dirichlet series interference suffices to obtain zeta. However, in order to create zeta along the critical line where the non-trivial zeros are located, we need two entangled quantum systems. In this way entanglement may be considered the quantum analogue of the analytical continuation of complex analysis. We also analyze the Newton flows of zeta as well as of the closely related function xi. Both provide additional insight into the Riemann hypothesis.
(Asked what he would first do, if he were magically awakened aftercenturies, David Hilbert replied "I would ask whetheranyone had proved the Riemann Hypothesis.")ζ(.) was defined for convergent cases in Euler's mini-bio,which Riemann extended via analytic continuation for all cases.
What is the analytic continuation of the Riemann Zeta …
A. The Riemann Hypothesis
All nontrivial zeros of the analytical continuation of the Riemann zeta function have a real part of 1/2. The states that the imaginary parts of the zeros of the Zeta function corresponds to eigenvalues of an unbounded self adjoint operator.
We provide a solution for the RH building on a new , alternatively to the current Gauss-Weierstrass function based Zeta function theory. This primarily enables a proof of the (but also of other RH criteria like the or Polya criteria), whereby the imaginary parts of the zeros of the corresponding alternative Zeta function definition corresponds to eigenvalues of a self adjoint operator with (newly) distributional Hilbert space domain.
Analytic Continuation and the Riemann Zeta Function | …
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