Some applications of Number Theoretical Universality

Number 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 finite-dimensional extensions of p-adic numbers (e^p is ordinary p-adic number). This is extremely powerful physics inspired conjecture with a wide range of possible mathematical applications.

1. For instance, vacuum functional defined as an exponent of Kähler action for preferred externals would be number of this kind. One could define functional integral as adelic operation in all number fields: essentially as sum of exponents of Kähler action for stationary preferred extremals since Gaussian and metric determinants potentially spoiling NTU would cancel each other leaving only the exponent.
   
   
2. The implications of NTU for the zeros of Riemann zeta expected to be closely related to super-symplectic conformal weights will be discussed below.
   
   
3. NTU generalises to all Lie groups. Exponents exp(in_iJⁱ/n) of lie-algebra generators define generalisations of number theoretically universal group elements and generate a discrete subgroup of compact Lie group. Also hyperbolic "phases" based on the roots e^m/n are possible and make possible discretized NTU versions of all Lie-groups expected to play a key role in adelization of TGD.
   

   NTU generalises also to quaternions and octonions and allows to define them as number theoretically universal entities. Note that ordinary p-adic variants of quaternions and octonions do not give rise to a number field: inverse of quaternion can have vanishing p-adic variant of norm squared satisfying ∑_n x_n²=0.
   

   NTU allows to define also the notion of Hilbert space as an adelic notion. The exponents of angles characterising unit vector of Hilbert space would correspond to roots of unity.
   
   

Super-symplectic 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 infinite-dimensional. Super-symplectic 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 n-multiples 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 non-vanishing 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.

1. Riemann Zeta ζ(s)= ∏_p(1/(1-p^-s) identifiable formally as a partition function ζ_B(s) of arithmetic boson gas with bosons with energy log(p) and temperature 1/s= 1/(1/2+iy) should be replaced with that of arithmetic fermionic gas given in the product representation by ζ_F(s) =∏_p (1+p^-s) so that the identity ζ_B(s))/ζ_F(s) =ζ_B(2s) follows. This gives
   

   ζ_B(s)/ζ_B(2s) .
   

   ζ_F(s) has zeros at zeros s_n of ζ (s) and at the pole s=1/2 of zeta(2s). ζ_F(s) has poles at zeros s_n/2 of ζ(2s) and at pole s=1 of ζ(s).
   

   The spectrum of 1/T would be for the generators of algebra {(-1/2+iy)/2, n>0, -1}. In p-adic thermodynamics the p-adic temperature is 1/T=1/n and corresponds to "trivial" poles of ζ_F. Complex values of temperature does not make sense in ordinary thermodynamics. In ZEO quantum theory can be regarded as a square root of thermodynamics and complex temperature parameter makes sense.
   
   

2. If the spectrum of conformal weights of generators of algebra (not the entire algebra!) corresponds to poles serving as analogs of propagator poles, it consists of the "trivial" conformal h=n>0- the standard spectrum with h=0 assignable to massless particles excluded - and "non-trivial" h=-1/4+iy/2. There is also a pole at h=-1.
   

   Both the non-trivial pole with real part h_R= -1/4 and the pole h=-1 correspond to tachyons. I have earlier proposed conformal confinement meaning that the total conformal weight for the state is real. If so, one obtains for a conformally confined two-particle states corresponding to conjugate non-trivial zeros in minimal situation h_R= -1/2 assignable to N-S representation.
   

   In p-adic mass calculations ground state conformal weight must be -5/2. The negative fermion ground state weight could explain why the ground state conformal weight must be tachyonic -5/2. With the required 5 tensor factors one would indeed obtain this with minimal conformal confinement. In fact, arbitrarily large tachyonic conformal weight is possible but physical state should always have conformal weights h>0.
   
   

3. h=0 is not possible for generators, which reminds of Higgs mechanism for which the naive ground states corresponds to tachyonic Higgs. h=0 conformally confined massless states are necessarily composites obtained by applying the generators of Kac-Moody algebra or super-symplectic algebra to the ground state. This is the case according to p-adic mass calculations, and would suggest that the negative ground state conformal weight can be associated with super-symplectic algebra and the remaining contribution comes from ordinary super-conformal generators. Hadronic masses whose origin is poorly understood could come from super-symplectic degrees of freedom. There is no need for p-adic thermodynamics in super-symplectic degrees of freedom.
   
   

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.

1. The numerical calculation of Fourier transform for the distribution of the imaginary parts iy of zeros s=1/2+iy of zeta shows that it is concentrated at discrete set of frequencies coming as log(pⁿ), p prime. This translates to the statement that the zeros of zeta form a 1-dimensional quasicrystal, a discrete structure Fourier spectrum by definition is also discrete (this of course holds for ordinary crystals as a special case). Also the logarithms of powers of primes would form a quasicrystal, which is very interesting from the point of view of p-adic length scale hypothesis. Primes label the "energies" of elementary fermions and bosons in arithmetic number theory, whose repeated second quantization gives rise to the hierarchy of infinite primes. The energies for general states are logarithms of integers.
   
   
2. Powers pⁿ label the points of quasicrystal defined by points log(pⁿ) and Riemann zeta has interpretation as partition function for boson case with this spectrum. Could pⁿ label also the points of the dual lattice defined by iy?
   
   
3. The existence of Fourier transform for points log(p_iⁿ) for any vector y_a requires p_i^iy_a to be a root of unity. This could define the sense in which zeros of zeta are universal. This condition also guarantees that the factor n^-1/2-iy appearing in zeta at critical line are number theoretically universal (p^1/2 is problematic for Q_p: the problem might be solved by eliminating from p-adic analog of zeta the factor 1-p^-s.
   
   1. One obtains for the pair (p_i,s_a) the condition log(p_i)y_a= q_ia2π, where q_ia is a rational number. Dividing the conditions for (i,a) and (j,a) gives
      

      p_i= p_j^q_ia/q_ja
      

      for every zero s_a so that the ratios q_ia/q_ja do not depend on s_a. Since the exponent is rational number one obtains p_i^M= p_j^N for some integers, which cannot be true.
      
      

   2. Dividing the conditions for (i,a) and (i,b) one obtains
      

      y_a/y_b= q_ia/q_ib
      

      so that the ratios q_ia/q_ib do not depend on p_i. The ratios of the imaginary parts of zeta would be therefore rational number which is very strong prediction and zeros could be mapped by scaling y_a/y₁ where y₁ is the zero which smallest imaginary part to rationals.
      
      

   3. The impossible consistency conditions for (i,a) and (j,a) can be avoided if each prime and its powers correspond to its own subset of zeros and these subsets of zeros are disjoint: one would have infinite union of sub-quasicrystals labelled by primes and each p-adic number field would correspond to its own subset of zeros: this might be seen as an abstract analog for the decomposition of rational to powers of primes. This decomposition would be natural if for ordinary complex numbers the contibution in the complement of this set to the Fourier trasform vanishes. The conditions (i,a) and (i,b) require now that the ratios of zeros are rationals only in the subset associated with p_i.
      

For the general option the Fourier transform can be delta function for x=log(p^k) and the
set {y_a(p)} contains N_p zeros. The following argument inspires the conjecture that
for each p there is an infinite number N_p of zeros y_a(p) satisfying

p^iy_a(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 y₀=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^iy_a(p_i) would fail to be stationary. The phase e^ixy would be non-stationary also for x≠ log(p^k) as function of y.

1. Assume that for x =qlog(p), q not a rational, the phases e^ixy fail to be roots of unity and are random implying the vanishing/smallness of F(x) .
   
   
2. Assume that for a given p all powers p^iy for y not in {y_a(p)} fail to be roots of unity and are also random so that the contribution of the set y not in {y_a(p)} to F(p) vanishes/is small.
   
   
3. For x= log(p^k/m) the Fourier transform should vanish or be small for m different from 1 (rational roots of primes) and give a non-vanishing contribution for m=1. One has
   

   F(x= log(p^k/m ) =∑_{1≤ n≤ N(p)} e^{[kM(n,p)/mN(n,p)]i2π} .
   

   Obviously one can always choose N(n,p)=N(p).
   
   

4. For the simplest option N(p)=1 one would obtain delta function distribution for x=log(p^k). The sum of the phases associated with y_a(p) and -y_a(p) from the half axes of the critical line would give
   

   F(x= log(pⁿ)) ∝ X(pⁿ)==2cos(n× (r(p)/m(p))× 2π) .
   

   The sign of F would vary.
   
   

5. The rational r(p)/m(p) would characterize given prime (one can require that r(p) and m(p) have no common divisors). F(x) is non-vanishing for all powers x=log(pⁿ) for m(p) odd. For p=2, also m(2)=2 allows to have |X(2ⁿ)|=2. An interesting ad hoc ansatz is m(p)=p or p^s(p). One has periodicity in n with period m(p) that is logarithmic wave. This periodicity serves as a test and in principle allows to deduce the value of r(p)/m(p) from the Fourier transform.
   
   

What could one conclude from the data (see this)?

1. The first graph gives |F(x=log(p^k| and second graph displays a zoomed up part of |F(x| for small powers of primes in the range [2,19]. For the first graph the eighth peak (p=11) is the largest one but in the zoomed graphs this is not the case. Hence something is wrong or the graphs correspond to different approximations suggesting that one should not take them too seriously.
   

   In any case, the modulus is not constant as function of p^k. For small values of p^k the envelope of the curve decreases and seems to approach constant for large values of p^k (one has x< 15 (e¹⁵≈ 3.3× 10⁶).
   
   

2. According to the first graph | F(x)| decreases for x=klog(p)<8, is largest for small primes, and remains below a fixed maximum for 8<x<15. According to the second graph the amplitude decreases for powers of a given prime (say p=2). Clearly, the small primes and their powers have much larger | F(x)| than large primes.
   
   

There are many possible reasons for this behavior. Most plausible reason is that the sums involved converge slowly and the approximation used is not good. The inclusion of only 10⁴ zeros would show the positions of peaks but would not allow reliable estimate for their intensities.

1. The distribution of zeros could be such that for small primes and their powers the number of zeros is large in the set of 10⁴ zeros considered. This would be the case if the distribution of zeros y_a(p) is fractal and gets "thinner" with p so that the number of contributing zeros scales down with p as a power of p, say 1/p, as suggested by the envelope in the first figure.
   
   
2. The infinite sum, which should vanish, converges only very slowly to zero. Consider the contribution Δ F(p^k,p₁) of zeros not belonging to the class p₁≠ p to F(x=log(p^k)) =∑_pi Δ F(p^k,p_i), which includes also p_i=p. Δ F(p^k,p_i), p≠ p₁ should vanish in exact calculation.
   
   
   1. By the proposed hypothesis this contribution reads as
      

      l Δ F(p,p₁)= ∑_a cos[X(p^k,p₁)(M(a,p₁)+ r(p₁)/m(p₁))2π)t] .

      X(p^k,p₁)=log(p^k)/log(p₁).
      

      Here a labels the zeros associated with p₁. If p^k is "approximately divisible" by p¹ in other words, p^k≈ np₁, the sum over finite number of terms gives a large contribution since interference effects are small, and a large number of terms are needed to give a nearly vanishing contribution suggested by the non-stationarity of the phase. This happens in several situations.
      
      

   2. The number π(x) of primes smaller than x goes asymptotically like π(x) ≈ x/log(x) and prime density approximately like 1/log(x)-1/log(x)² so that the problem is worst for the small primes. The problematic situation is encountered most often for powers p^k of small primes p near larger prime and primes p (also large) near a power of small prime (the envelope of | F(x)| seems to become constant above x∼ 10³).
      
      
   3. The worst situation is encountered for p=2 and p₁=2^k-1 - a Mersenne prime and p₁= 2^2k+1, k≤ 4 - Fermat prime. For (p,p₁)=(2^k,M_k) one encounters X(2^k,M_k)= (log(2^k)/log(2^k-1) factor very near to unity for large Mersennes primes. For (p,p₁)=(M_k,2) one encounters X(M_k,2)= (log(2^k-1)/log(2) ≈ k. Examples of Mersennes and Fermats are (3,2),(5,2),(7,2),(17,2),(31,2), (127,2),(257,2),... Powers 2^k, k=2,3,4,5,7,8,.. are also problematic.
      
      
   4. Also twin primes are problematic since in this case one has factor X(p=p₁+2,p₁)=log(p₁+2)/log(p₁). The region of small primes contains many twin prime pairs: (3,5), (5,7), (11,13), (17,19), (29,31),....
      
      
   These observations suggest that the problems might be understood as resulting from including too small number of zeros.
3. The predicted periodicity of the distribution with respect to the exponent k of p^k is not consistent with the graph for small values of prime unless the periodic m(p) for small primes is large enough. The above mentioned effects can quite well mask the periodicity. If the first graph is taken at face value for small primes, r(p)/m(p) is near zero, and m(p) is so large that the periodicity does not become manifest for small primes. For p=2 this would require m(2)>21 since the largest power 2ⁿ≈ e¹⁵ corresponds to n∼ 21.
   
   

To summarize, the prediction is that for zeros of zeta should divide into disjoint classes {y_a(p)\ labelled by primes such that within the class labelled by p one has p^iy_a(p)=e^{(r(p)/m(p))i2π} so that has y_a(p) = [M(a,p) +r(p)/m(p))] 2π/log(p).

What this speculative picture from the point of view of TGD?

1. A possible formulation for number theoretic universality for the poles of fermionic Riemann zeta ζ_F(s)= ζ(s)/ζ(2s) could be as a condition that is that the exponents p^ks_n(p)/2= p^k/4p^iky_n(p)/2 exist in a number theoretically universal manner for the zeros s_n(p) for given p-adic prime p and for some subset of integers k. If the proposed conditions hold true, exponent reduces p^k/4 requiring that k is a multiple of 4. The number of the non-trivial generating elements of super-symplectic algebra in the monomial creating physical state would be a multiple of 4. These monomials would have real part of conformal weight -1. Conformal confinement suggests that these monomials are products of pairs of generators for which imaginary parts cancel. The conformal weights are however effectively real for the exponents automatically. Could the exponential formulation of the number theoretic universality effectively reduce the generating elements to those with conformal weight -1/4 and make the operators in question hermitian?
   
   
2. Quasi-crystal property might have an application to TGD. The functions of light-like radial coordinate appearing in the generators of supersymplectic algebra could be of form r^s, s zero of zeta or rather, its imaginary part. The eigenstate property with respect to the radial scaling rd/dr is natural by radial conformal invariance.
   

   The idea that arithmetic QFT assignable to infinite primes is behind the scenes in turn suggests light-like momenta assignable to the radial coordinate have energies with the dual spectrum log(pⁿ). This is also suggested by the interpretation of ζ as square root of thermodynamical partition function for boson gas with momentum log(p) and analogous interpretation of ζ_F.
   

   The two spectra would be associated with radial scalings and with light-like translations of light-cone boundary respecting the direction and light-likeness of the light-like radial vector. log(pⁿ) spectrum would be associated with light-like momenta whereas p-adic mass scales would characterize states with thermal mass. Note that generalization of p-adic length scale hypothesis raises the scales defined by pⁿ to a special physical position: this might relate to ideal structure of adeles.
   
   

3. Finite measurement resolution suggests that the approximations of Fourier transforms over the distribution of zeros taking into account only a finite number of zeros might have a physical meaning. This might provide additional understand about the origins of generalized p-adic length scale hypothesis stating that primes p≈ p₁^k, p₁ small prime - say Mersenne primes - have a special physical role.
   
   

See the chapter Unified Number Theoretic Vision of "TGD as Generalized Number Theory" or the article Could one realize number theoretical universality for functional integral?.

For a summary of earlier postings see Links to the latest progress in TGD.