Farey sequences, Riemann Hypothesis, and Platonia as the best possible world

Kea has mentioned Farey sequences in her blog couple of times (see this and this).

Some basic facts about Farey sequences demonstrate that they are very interesting also from TGD point of view.

1. Farey sequence F_N is defined as the set of rationals 0< q= m/n≤1 satisfying the conditions n≤ N ordered in an increasing sequence.

2. Two subsequent terms a/b and c/d in F_N satisfy the condition ad-bc=1 and thus define and element of the modular group SL(2,Z).

3. The number of terms in Farey sequence is given by

   F(N) =F(N-1)+ φ(N-1).

   Here φ(n) is Euler's totient function giving the number of divisors of n. For primes one has φ(p)=1 so that in the transition from p to p+1 the length of Farey sequence increases by one unit by the addition of q=1/(p+1) to the sequence.

The members of Farey sequence F_N are in one-one correspondence with the set of quantum phases q_n=exp(i2π/n) , 0≤ n≤ N. This suggests a close connection with the hierarchy of Jones inclusions, quantum groups, and in TGD context with quantum measurement theory with finite measurement resolutiona nd the hierarchy of Planck constants involving the generalization of imbedding space. Also the recent TGD inspired ideas about the hierachy of subgroups of rational modular group with subgroups labelled by N and in direct correspondence with the hierarchy of quantum critical phases would naturally relate to Farey sequence.

1. Riemann Hypothesis and Farey sequences

Farey sequences are used in two equivalent formulations of the Riemann hypothesis. Suppose the terms of F_N are a_n,N, 0 < n≤ m_N. Define d_n,N = a_n,N - n /m_N: in other words d_n,N is the difference between the n:th term of the N:th Farey sequence, and the n:th member of a set of the same number of points, distributed evenly on the unit interval. Franel and Landau proved that both of the two statements

• ∑_n=1,...,mNd_n,N =O(N^r) for any r>1/2.

• ∑_n=1,...,mN d_n,N² =O(N^r) for any r>1.

are equivalent with Riemann hypothesis.

One can say that RH would guarantee that the numbers of Farey sequence provide the best possible approximate representation for the evenly distributed rational numbers n/m_N.

2. Farey sequences and TGD

Farey sequences seem to relate very closely to TGD.

1. The rationals in the Farey sequence can be mapped to the roots of unity by the map q→exp(i2π q). The numbers 1/m_N are in turn mapped to the numbers exp(i2π/m_N), which are also roots of unity. The statement would be that the algebraic phases defined by Farey sequence give the best possible approximate representation for the phases exp(in2π/m_N) with evenly distributed phase angle.

2. In TGD framework the phase factors defined by F_N corresponds to the set of quantum phases corresponding to Jones inclusions labelled by q=exp(i2π/n), n≤ N, and thus to the N lowest levels of dark matter hierarchy. There are actually two hierarchies corresponding to M⁴ and CP₂ degrees of freedom and the Planck constant appearing in Schrödinger equation corresponds to the ratio n_a/n_b defining quantum phases in these degrees of freedom. Z_{na× nb} appears as a conformal symmetry of "dark" partonic 2-surfaces and with very general assumptions this implies that there are only three fermion families in TGD Universe.

3. The fusion of physics associated with various number fields to single coherent whole requires algebraic universality. In particular, the roots of unity, which are complex algebraic numbers, should define approximations to continuum of phase factors. At least the S-matrix associated with p-adic-to-real transitions and more generally p₁ → p₂ transitions between states for which the partonic space-time sheets are p₁- resp. p₂-adic can involve only this kind of algebraic phases. One can also say that cognitive representations can involve only algebraic phases and algebraic numbers in general. For real-to-real transitions and real-to-padic transitions U-matrix might be non-algebraic or obtained by analytic continuation of algebraic U-matrix. S-matrix is by definition diagonal with respect to number field and similar continuation principle might apply also in this case.

4. The subgroups of the hierarchy of subgroups of the modular group with rational matrix elements are labelled by integer N and relate naturally to the hierarchy of Farey sequences. The hierarchy of quantum critical phases is labelled by integers N with quantum phase transitions occuring only between phases for which the smaller integer divides the larger one.

5. The 2-tangles known as rational tangles form are characterized by a rational number a/b (for detailed definitions see the article of Kaufmann and Lambropoulou). According to the result of the same article, two rational tangles labelled by a/b and c/d and possessing commutative sum and product combine to form an unknot if and only if a/b and c/d are two subsequent Farey numbers and therefore satisfy ad-bc=+/-1. An interesting question is whether the result somehow generalizes to the case of N-tangles and whether this generalization relates to the hierarchy of subgroups of the rational modular group obtained by replacing the generator τ→τ+1 with τ→ τ+1/N.

3. Interpretation of RH in TGD framework

Number theoretic universality of physics suggests an interpretation for the Riemann hypothesis in TGD framework. RH would be equivalent to the statement that the Farey numbers provide best possible approximation to the set of rationals k/m_N. Or to the statement that the roots of unity contained by F_N define the best possible approximation for the roots of unity defined as exp(ik2π/m_N) with evenly spaced phase angles. The roots of unity allowed by the lowest N levels of the hierarchy of Jones inclusions allows the best possible approximate representation for algebraic phases represented exactly at m_N:th level of hierarchy.

A stronger statement would be that the Platonia where RH holds true would be the best possible world in the sense that algebraic physics behind the cognitive representations would allow the best possible approximation hierarchy for the continuum physics (both for numbers in unit interval and for phases on unit circle). Platonia with RH would be cognitive paradise;-).

One could see this also from different view point. "Platonia as the cognitively best possible world" could be taken as the "axiom of all axioms": a kind of fundamental variational principle of mathematics. Among other things it would allow to conclude that RH is true: RH must hold true either as a theorem following from some axiomatics or as an axiom in itself.

For details see the chapter Hyper-Finite Factors and Construction of S-Matrix of "Towards S-Matrix".