Does Hilbert-Polya Conjecture Infer Riemann Hypothesis by the means of spectral theory? Can anyone describe the procedure
Yes, but the problem then is to find the appropriate operator. A summary may be found here: https://en.wikipedia.org/wiki/Hilbert%E2%80%93P%C3%B3lya_conjecture
See also this: https://arxiv.org/pdf/cond-mat/9712010.pdf
However it's not necessary for the Hilbert-Polya conjecture to be true, in order for the Riemann Hypothesis be true.
Stam Nicolis , you said that
" However it's not necessary for the Hilbert-Polya conjecture to be true, in order for the Riemann Hypothesis to be true. "
Why is that?
Because the existence of such a Hermitian operator isn't imposed by anything else. If such an operator exists, the Riemann hypothesis is true; but such an operator need not exist in the first place. The Riemann hypothesis can be true, even if such an operator doesn't exist. Indeed, since it's not known how to produce all the prime numbers-and only the prime numbers-it's not known how to construct such an operator. Indeed to construct it in the first place it would be necessary to prove the Riemann hypothesis independently, since there are constraints on the sequence of eigenvalues of a Hermitian operator-which are discussed in the paper I linked to. And checking these constraints amounts to proving the Riemann hypothesis.
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