Monday, September 28, 2015

Are the zeros of Riemann zeta number theoretically universal?

I have already posted the following piece of text as part of earlier posting. Since the outcome of simple argument leads to very powerful statement about number theoretic anatomy of
the zeros of Riemann zeta, I thought that it would be approriate to represent it separately.

Dyson's comment about Fourier transform of Riemann Zeta is very interesting concerning Number Theoretic Universality (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(pin) for any vector ya requires piiya 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 (p1/2 is problematic for Qp: the problem might be solved by eliminating from p-adic analog of zeta the factor 1-p-s.

    1. One obtains for the pair (pi,sa) the condition log(pi)ya= qia2π, where qia is a rational number. Dividing the conditions for (i,a) and (j,a) gives

      pi= pjqia/qja

      for every zero sa so that the ratios qia/qja do not depend on sa. Since the exponent is rational number one obtains piM= pjN for some integers, which cannot be true.

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

      ya/yb= qia/qib

      so that the ratios qia/qib do not depend on pi. 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 ya/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 pi.
For the general option the Fourier transform can be delta function for x=log(pk) and the set {ya(p)} contains Np zeros. The following argument inspires the conjecture that for each p there is an infinite number Np of zeros ya(p) satisfying

piya(p)=u(p)=e(r(p)/m(p))i2π ,

where u(p) is a root of unity that is ya(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 ya(p) associated with p would have constant stationary phase whereas for ya(pi≠ p)) the phase piya(pi) would fail to be stationary. The phase eixy would be non-stationary also for x≠ log(pk) as function of y.

  1. Assume that for x =qlog(p), q not a rational, the phases eixy 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 piy for y not in {ya(p)} fail to be roots of unity and are also random so that the contribution of the set y not in {ya(p)} to F(p) vanishes/is small.

  3. For x= log(pk/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(pk/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(pk). The sum of the phases associated with ya(p) and -ya(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 ps(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(pk| 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 pk. For small values of pk the envelope of the curve decreases and seems to approach constant for large values of pk (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 ya(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(pk,p1) of zeros not belonging to the class p1\ne p to F(x=log(pk)) =∑pi Δ F(pk,pi), which includes also pi=p. Δ F(pk,pi), p≠ p1 should vanish in exact calculation.

    1. By the proposed hypothesis this contribution reads as

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


      Here a labels the zeros associated with p1. If pk is "approximately divisible" by p1 in other words, pk≈ 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 pk 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=2k-1 - a Mersenne prime and p1= 22k+1, k≤ 4 - Fermat prime. For (p,p1)=(2k,Mk) one encounters X(2k,Mk)= (log(2k)/log(2k-1) factor very near to unity for large Mersennes primes. For (p,p1)=(Mk,2) one encounters X(Mk,2)= (log(2k-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 2k, 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 pk 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 {ya(p)\ labelled by primes such that within the class labelled by p one has piya(p)=e(r(p)/m(p))i2π so that has ya(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 ζF(s)= ζ(s)/ζ(2s) could be as a condition that is that the exponents pksn(p)/2= pk/4pikyn(p)/2 exist in a number theoretically universal manner for the zeros sn(p) for given p-adic prime p and for some subset of integers k. If the proposed conditions hold true, exponent reduces pk/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 rs, 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 ζF.

    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≈ p1k, 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.


Anonymous said... said...

Interesting. Compression, expansion, compression... Penrose's argument was that quasicrystals cannot grow little by little but emerge as a whole via kind of macrocopic quantum phase transition. In TGD framework quantum criticality makes possible large values of h_eff=n*h. Quantum criticality could be perhaps associated with the collisions and create large h_eff phase transforming to a phase characterize by p-adic length scale which is same as the quantum scale associated with large h_eff. I have proposed that this happens in biology.