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