Riemann hypothesis and quantum TGD

The basic mathematical structure of quantum TGD lead for year or so ago to a sharpening of Riemann hypothesis. The zeros of Riemann Zeta are of form x=1/2+iy and p^(iy) is rational phase for every prime and thus defines Pythagorean triangle (orthogonal triangle with integer-valued sides).

One important role of p-adic numbers is related to the determination of the zeros of polynomials. This suggest an obvious strategy of proof of the sharpened Riemann hypothesis.

a) Continue Riemann zeta for every value of p to p-adic Zeta function defined in algebraic extension allowing square root of all ordinary p-adic numbers. p-Adic Zetas are defined in the set x=n/2+ iy, where n is integer and y defines Pythagorean triangle.

b) Prove a generalization of 'Local-Global principle' for Diophantine equation stating that the existence or non-existence of solutions in the set of rationals can be detected by studying, for each prime, the solutions of the equation in p-adic number fields. This principle is too good to be true generally but there are some simple situations when this the case. For instance, Hasse-Minkowski theorem