Riemann Hypothesis and Physics

Riemann hypothesis states that the nontrivial zeros of Riemann Zeta function lie on the axis x = 1/2. Since Riemann zeta function allows interpretation as a thermodynamical partition function for a quantum field theoretical system consisting of bosons labelled by primes, it is interesting to look Riemann hypothesis from the perspective of physics. Quantum TGD and also TGD inspired theory of consciousness provide additional view points to the hypothesis and suggests sharpening of Riemann hypothesis, detailed strategies of proof of the sharpened hypothesis, and heuristic arguments for why the hypothesis is true. The idea that the evolution of cognition involves the increase of the dimensions of finite-dimensional extensions of p-adic numbers associated with p-adic space-time sheets emerges naturally in TGD inspired theory of consciousness. A further input that led to a connection with Riemann Zeta was the work of Hardmuth Mueller [34] suggesting strongly that e and its p − 1 powers at least should belong to the extensions of p-adics. The basic objects in Mueller’s approach are so called logarithmic waves exp(iklog(u)) which should exist for u = n for a suitable choice of the scaling momenta k. Logarithmic waves appear also as the basic building blocks (the terms n = exp(log(n)(Re[s]+iIm[s])) in Riemann Zeta. This inspires naturally the hypothesis that also Riemann Zeta function is universal in the sense that it is defined at is zeros s = 1/2+ iy not only for complex numbers but also for all p-adic number fields provided that an appropriate finite-dimensional extensions involving also transcendentals are allowed. This allows in turn to algebraically continue Zeta to any number field. The zeros of Riemann zeta are determined by number theoretical quantization and are thus universal and should appear in the physics of critical systems. The hypothesis log(p) = q1(p)exp[q2(p)] π explains the length scale hierarchies based on powers of e, primes p and Golden Mean. Mueller’s logarithmic waves lead also to an elegant concretization of the Hilbert Polya conjecture and to a sharpened form of Riemann hypothesis: the phases q−iy for the zeros of Riemann Zeta belong to a finite-dimensional extension of Rp for any value of primes q and p and any zero 1/2 + iy of ζ. The question whether the imaginary parts of the Riemann Zeta are linearly independent (as assumed in the previous work) or not is of crucial physical significance. Linear independence implies that the spectrum of the super-canonical weights is essentially an infinite-dimensional lattice. Otherwise a more complex structure results. The numerical evidence supporting the translational invariance of the correlations for the spectrum of zeros together with padic considerations leads to the working hypothesis that for any prime p one can express the spectrum of zeros as the product of p powers