Abstract

The proper mathematical proof of Riemann's hypothesis (RH) in Hilbert mathematics is suggested. It follows the methodological and philosophical considerations in Part I of the paper. Riemann's zeta function is continued "physically" at its single and simple pole conventionally to be square integrable there (though not being analytical only there) and thus everywhere on the complex plane in order to be interpreted as a wave function (though with a singularity at the pole, and thus generalizing the Hilbert-Polya conjecture's viewpoint). Therefore, is interpreted physically corresponding to ζ(????) some quantum state described by it This allows for applying the Noether (1918) first theorem after its reformulation by means of the newly introduced "nonstandard bijection" meaning the mapping of any mathematical structure and its nonstandard model after the Löwenheim-Skolem theorem. Then, the additive semigroup of its trivial zeros implies for all nontrivial zeros to share the critical line (i.e. proving RH) furthermore reasoning the reciprocity of the former generator "2" and the constant "½" of "Re(s) = ½". That approach for proving RH implies as "byproducts" also: the fundamentally random (GUE) distribution of all nontrivial zeros due to "nonstandard bijection"; the generalization of the axiom of induction to a conservation low as for the set-theoretical "continuation" of arithmetic (thus from the axiom of induction the axiom of infinity); the generalization of the Noether theorem in a few consecutive levels reaching even to set theory and the foundations of mathematics (after Hilbert arithmetic and ontomathematics). An idea for proving Goldbach's conjecture in Hilbert arithmetic (in a forthcoming paper) is reasoned as a corollary from the same approach for proving RH.