What's new inPhysics as a Generalized Number TheoryNote: Newest contributions are at the top! 
Year 2015 
Why the nontrivial zeros of Riemann zeta should reside at critical line?Riemann Hypothess (RH) states that the nontrivial (critical) zeros of zeta lie at critical line s=1/2. It would be interesting to know how many physical justifications for why this should be the case has been proposed during years. Probably this number is finite, but very large it certainly is. In Zero Energy Ontology (ZEO) forming one of the cornerstones of the ontology of quantum TGD, the following justification emerges naturally. I represented it in the answer to previous posting, but there was stupid error in the answer so that I represent the corrected argument here.

Does Riemann Zeta Code for Generic Coupling Constant Evolution?The understanding of coupling constant evolution and predicting it is one of the greatest challenges of TGD. During years I have made several attempts to understand coupling evolution.
See the new chapter Does Riemann Zeta Code for Generic Coupling Constant Evolution? or the article Does Riemann Zeta Code for Generic Coupling Constant Evolution?. 
Some applications of Number Theoretic UniversalityNumber Theoretic Universality (NTU) in the strongest form says that all numbers involved at "basic level" (whatever this means!) of adelic TGD are products of roots of unity and of power of a root of e defining finitedimensional extensions of padic numbers (e^{p} is ordinary padic number). This is extremely powerful physics inspired conjecture with a wide range of possible mathematical applications.
Supersymplectic conformal weights and Riemann zeta The existence of WCW geometry highly nontrivial already in the case of loop spaces. Maximal group of isometries required and is infinitedimensional. Supersymplectic algebra is excellent candidate for the isometry algebra. There is also extended conformal algebra associated with δ CD. These algebras have fractal structure. Conformal weights for isomorphic subalgebra nmultiples of those for entire algebra. Infinite hierarchy labelled by integer n>0. Generating conformal weights could be poles of fermionic zeta ζ_{F}. This demands n>0. Infinite number of generators with different nonvanishing conformal weight with other quantum numbers fixed. For ordinary conformal algebras there are only finite number of generating elements (n=1). If the radial conformal weights for the generators of g consist of poles of ζ_{F}, the situation changes. ζ_{F} is suggested by the observation that fermions are the only fundamental particles in TGD.
Are the zeros of Riemann zeta number theoretically universal? Dyson's comment about Fourier transform of Riemann Zeta is very interesting concerning NTU for Riemann zeta.
p^{iya(p)}=u(p)=e^{(r(p)/m(p))i2π} , where u(p) is a root of unity that is y_{a}(p)=2π (m(a)+r(p))/log(p) and forming a subset of a lattice with a lattice constant y_{0}=2π/log(p), which itself need not be a zero. In terms of stationary phase approximation the zeros y_{a}(p) associated with p would have constant stationary phase whereas for y_{a}(p_{i}≠ p)) the phase p^{iya(pi)} would fail to be stationary. The phase e^{ixy} would be nonstationary also for x≠ log(p^{k}) as function of y.
What this speculative picture from the point of view of TGD?

More about physical interpretation of algebraic extensions of rationalsThe number theoretic vision has begun to show its power. The basic hierarchies of quantum TGD would reduce to a hierarchy of algebraic extensions of rationals and the parameters  such as the degrees of the irreducible polynomials characterizing the extension and the set of ramified primes  would characterize quantum criticality and the physics of dark matter as large h_{eff} phases. The identification of preferred padic primes as remified primes of the extension and generalization of padic length scale hypothesis as prediction of NMP are basic victories of this vision (see this and this). By strong form of holography the parameters characterizing string world sheets and partonic 2surfaces serve as WCW coordinates. By various conformal invariances, one expects that the parameters correspond to conformal moduli, which means a huge simplification of quantum TGD since the mathematical apparatus of superstring theories becomes available and number theoretical vision can be realized. Scattering amplitudes can be constructed for a given algebraic extension and continued to various number fields by continuing the parameters which are conformal moduli and group invariants characterizing incoming particles. There are many unanswered and even unasked questions.
1. Some basic notions Some basic facts about extensions are in order. I emphasize that I am not a specialist. 1.1. Basic facts The algebraic extensions of rationals are determined by roots of polynomials. Polynomials be decomposed to products of irreducible polynomials, which by definition do not contain factors which are polynomials with rational coefficients. These polynomials are characterized by their degree n, which is the most important parameter characterizing the algebraic extension. One can assign to the extension primes and integers  or more precisely, prime and integer ideals. Integer ideals correspond to roots of monic polynomials P_{n}(x)=x^{n}+..a_{0} in the extension with integer coefficients. Clearly, for n=0 (trivial extension) one obtains ordinary integers. Primes as such are not a useful concept since roots of unity are possible and primes which differ by a multiplication by a root of unity are equivalent. It is better to speak about prime ideals rather than primes. Rational prime p can be decomposed to product of powers of primes of extension and if some power is higher than one, the prime is said to be ramified and the exponent is called ramification index. Eisenstein's criterion states that any polynomial P_{n}(x)= a_{n}x^{n}+a_{n1}x^{n1}+...a_{1}x+ a_{0} for which the coefficients a_{i}, i<n are divisible by p and a_{0} is not divisible by p^{2} allows p as a maximally ramified prime. mThe corresponding prime ideal is n:th power of the prime ideal of the extensions (roughly n:th root of p). This allows to construct endless variety of algebraic extensions having given primes as ramified primes. Ramification is analogous to criticality. When the gradient potential function V(x) depending on parameters has multiple roots, the potential function becomes proportional a higher power of xx_{0}. The appearance of power is analogous to appearance of higher power of prime of extension in ramification. This gives rise to cusp catastrophe. In fact, ramification is expected to be number theoretical correlate for the quantum criticality in TGD framework. What this precisely means at the level of spacetime surfaces, is the question. 1.2 Galois group as symmetry group of algebraic physics I have proposed long time ago that Galois group acts as fundamental symmetry group of quantum TGD and even made clumsy attempt to make this idea more precise in terms of the notion of number theoretic braid. It seems that this notion is too primitive: the action of Galois group must be realized at more abstract level and WCW provides this level. First some facts (I am not a number theory professional, as the professional reader might have already noticed!).
Galois group acts in the space of integers or prime ideals of the algebraic extension of rationals and it is also physically attractive to consider the orbits defined by ideals as preferred geometric structures. If the numbers of the extension serve as parameters characterizing string world sheets and partonic 2surfaces, then the ideals would naturally define subsets of the parameter space in which Galois group would act. The action of Galois group would leave the spacetime surface invariant if the sheets coincide at ends but permute the sheets. Of course, the spacetime sheets permuted by Galois group need not coincide at ends. In this case the action need not be gauge action and one could have nontrivial representations of the Galois group. In Langlands correspondence these representation relate to the representations of Lie group and something similar might take place in TGD as I have indeed proposed. Remark: Strong form of holography supports also the vision about quaternionic generalization of conformal invariance implying that the adelic spacetime surface can be constructed from the data associated with functions of two complex variables, which in turn reduce to functions of single variable. If this picture is correct, it is possible to talk about quantum amplitudes in the space defined by the numbers of extension and restrict the consideration to prime ideals or more general integer ideals.
2. How new degrees of freedom emerge for ramified primes? How the new discrete degrees of freedom appear for ramified primes?
3. About the physical interpretation of the parameters characterizing algebraic extension of rationals in TGD framework It seems that Galois group is naturally associated with the hierarchy h_{eff}/h=n of effective Planck constants defined by the hierarchy of quantum criticalities. n would naturally define the maximal order for the element of Galois group. The analog of singular covering with that of z^{1/n} would suggest that Galois group is very closely related to the conformal symmetries and its action induces permutations of the sheets of the covering of spacetime surface. Without any additional assumptions the values of n and ramified primes are completely independent so that the conjecture that the magnetic flux tube connecting the wormhole contacts associated with elementary particles would not correspond to very large n having the padic prime p characterizing particle as factor (p=M_{127}=2^{127}1 for electron). This would not induce any catastrophic changes. TGD based physics could however change the situation and reduce number theoretical degrees of freedom: the intuitive hypothesis that p divides n might hold true after all.

What could be the origin of padic length scale hypothesis?The argument would explain the existence of preferred padic primes. It does not yet explain padic length scale hypothesis stating that padic primes near powers of 2 are favored. A possible generalization of this hypothesis is that primes near powers of prime are favored. There indeed exists evidence for the realization of 3adic time scale hierarchies in living matter (see this) and in music both 2adicity and 3adicity could be present, this is discussed in TGD inspired theory of music harmony and genetic code (see this). The weak form of NMP might come in rescue here.
See the chapter Unified Number Theoretic Vision or the article The Origin of Preferred pAdic Primes?. 
What is the origin of the preferred padic primes?A longstanding question has been the origin of preferred padic primes characterizing elementary particles. I have proposed several explanations and the most convincing hitherto is related to the algebraic extensions of rationals and padic numbers selecting naturally preferred primes as those which are ramified for the extension in question. See the chapter Unified Number Theoretic Vision . 
About the twistorial description of lightlikeness in 8D sense using octonionic spinorsThe twistor approach to TGD require that the expression of lightlikeness of M^{4} momenta in terms of twistors generalizes to 8D case. The lightlikeness condition for twistors states that the 2× 2 matrix representing M^{4} momentum annihilates a 2spinor defining the second half of the twistor. The determinant of the matrix reduces to momentum squared and its vanishing implies the lightlikeness. This should be generalized to a situation in one has M^{4} and CP_{2} twistor, which are not lightlike separately but lightlikeness in 8D sense holds true (allowing massive particles in M^{4} sense and thus generalization of twistor approach for massive particles). The case of M^{8}=M^{4}× E^{4} M^{8}H duality suggests that it might be useful to consider first the twistorialiation of 8D lightlikeness first the simpler case of M^{8} for which CP_{2} corresponds to E^{4}. It turns out that octonionic representation of gamma matrices provide the most promising formulation. In order to obtain quadratic dispersion relation, one must have 2× 2 matrix unless the determinant for the 4× 4 matrix reduces to the square of the generalized lightlikeness condition.
The case of M^{8}=M^{4}× CP_{2} What about twistorialization in the case of M^{4}× CP_{2}? The introduction of wave functions in the twistor space of CP_{2} seems to be enough to generalize Witten's construction to TGD framework and that algebraic variant of twistors might be needed only to realize quantum classical correspondence. It should correspond to tangent space counterpart of the induced twistor structure of spacetime surface, which should reduce effectively to 4D one by quaternionicity of the spacetime surface.
To sum up, the generalization of the notion of twistor to 8D context allows description of massive particles using twistors but requires that octonionic Dirac equation is introduced. If one requires that octonionic and ordinary description of Dirac equation are equivalent, the description is possible only at surfaces having at most 1D CP_{2} projection  geodesic circle for the most stringent option. The boundaries of string world sheets are such surfaces and also string world sheets themselves if they have 1D CP_{2} projection, which must be geodesic circle if also induce gauge potentials are required to vanish. In spirit with M^{8}H duality, string boundaries give rise to classical M^{8} twistorizalization analogous to the standard M^{4} twistorialization and generalize 4momentum to massless 8momentum whereas imbedding space spinor harmonics give description in terms of fourmomentum and color quantum numbers. One has SO(4)SU(3) duality: a wave function in the space of 8momenta corresponds to SO(4) description of hadrons at low energies as opposed to that for quarks at high energies in terms of color. The M^{4} projection of the 8D M^{8} momentum must by quantum classical correspondence be equal to the fourmomentum assignable to imbedding spacespinor harmonics serving as building bricks for various superconformal representations. This is nothing but Equivalence Principle (EP) in the most concrete form: gravitational fourmomentum equals to inertial fourmomentum. EP for internal quantum numbers is clearly more delicate. In twistorialization also helicity is brought and for CP_{2} degrees of freedom M^{8} helicity means that electroweak spin is described in terms of helicity. Biologists have a principle known as "ontogeny recapitulates phylogeny" (ORP) stating that the morphogenesis of the individual reflects evolution of the species. The principle seems to be realized also in theoretical physics  at least in TGD Universe. ORP would now say that the evolution of theoretical physics via the emergence of increasingly complex notion of particle reflects the structure physics itself. Point like particles are really there as points at partonic 2surfaces carrying fermion number: their 1D orbits correspond to the boundaries of string world sheets; 2D hypercomplex string world sheets in flat space (M^{4}× S^{1}) are there and carry induced spinors; also complex (or cocomplex) partonic 2surfaces (Euclidian string world sheets) and carry particle numbers; 3D spacelike surfaces at the ends of causal diamonds (CDs) and the 3D lightlike orbits of partonic 2surfaces are there; 4D spacetime surfaces are there as quaternionic or coquaternionic submanifolds of 8D octonionic imbedding space: there the hierarchy ends since there are no higherdimensional classical number fields. ORP would thus also realize evolution of mathematics at the level of physics. The M^{4} projection of the 8D M^{8} momentum must by quantum classical correspondence be equal to the fourmomentum assignable to imbedding spacespinor harmonics serving as building bricks for various superconformal representations. This is nothing but Equivalence Principle in the most concrete form: gravitational fourmomentum equals to inertial fourmomentum. See the chapter TGD as a Generalized Number Theory II: Quaternions, Octonions, and their Hyper Counterparts or the article Classical part of the twistor story. 