Riemann Hypothesis and quasicrystals

Freeman Dyson has represented a highly interesting speculation related to Riemann hypothesis and 1-dimensional quasicrystals (QCs). He discusses QCs and Riemann hypothesis briefly in his Einstein lecture.

Dyson begins from the defining property of QC as discrete set of points of Euclidian space for which the spectrum of wave vectors associated with the Fourier transform is also discrete. What this says is that quasicrystal as also ordinary crystal creates discrete diffraction spectrum. This presumably holds true also in higher dimensions than D=1 although Dyson considers mostly D=1 case. Thus QC and its dual would correspond to discrete points sets. I will consider the consequences in TGD framework below.

Dyson considers first QCs at general level. Dyson claims that QCs are possible only in dimensions D=1,2,3. I do not know whether this is really the case. In dimension D=3 the known QCs have icosahedral symmetry and there are only very few of them. In 2-D case (Penrose tilings) there is n-fold symmetry, roughly one kind of QC associated with any regular polygon. Penrose tilings correspond to n=5. In 1-D case there is no point group (subgroup of rotation group) and this explains why the number of QCs is infinite. For instance, so called PV numbers identified as algebraic integers, which are roots of any polynomial with integer coefficients such that all other roots have modulus smaller than unity. 1-D QCs is at least as rich a structure as PV numbers and probably much richer.

Dyson suggests that Riemann hypothesis and its generalisations might be proved by studying 1-D quasi-crystals.

1. If Riemann Hypothesis is true, the spectrum for the Fourier transform of the distribution of zeros of Riemann zeta is discrete. The calculations of Andrew Odlycko indeed demonstrate this numerically, which is of course not a proof. From Dyson's explanation I understand that it consists of sums of integer multiples nlog(p) of logarithms of primes meaning that the non-vanishing Fourier components are apart from overall delta function (number of zeros) proportional to

   F(n)= ∑_sk n^-is_k=ζ_D(is_k) , s_k=1/2+iy_k ,

   where s_k are zeros of Zeta. ζ_D could be called the dual of zeta with summation over integers replaced with summation over zeros. For other "energies" than E=log(n) the Fourier transform would vanish. One can say that the zeros of Riemann Zeta and primes (or p-adic "energy" spectrum) are dual. Dyson conjectures that each generalized zeta function (or rather, L-function) corresponds to one particular 1-D QC and that Riemann zeta corresponds to one very special 1-D QC.

1. What is interesting that the same "energy" spectrum (logarithms of positive integers) appears in an arithmetic quantum field theory assignable to what I call infinite primes. An infinite hierarchy of second quantizations of ordinary arithmetic QFT is involved. A the lowest level the Fourier transform of the spectrum of the arithmetic QFT would consist of zeros of zeta rotated by π/2! The algebraic extensions of rationals and the algebraic integers associated with them define an infinite series of infinite primes and also generalized zeta functions obtained by the generalization of the sum formula. This would suggest a very deep connection with zeta functions, quantum physics, and quasicrystals. These zeta functions could correspond to 1-D QCs.
2. The definition of p-adic manifold (in TGD framework) forces a discretisation of M⁴× CP₂ having interpretation in terms of finite measurement resolution. This discretization induces also dicretization of space-time surfaces by induction of manifold structure. The discretisation of M⁴ (or E³) is achieved by crystal lattices, by QCs, and perhaps also by more general discrete structures. Could lattices and QCs be forced by the condition that the lattice like structures defines a discrete distributions with discrete spectrum? But why this?
3. There is also another problem. Integration is a problematic notion in p-adic context and it has turned out that discretization is unavoidable and also natural in finite measurement resolution. The inverse of the Fourier transform however involves integration unless the spectrum of the Fourier transform is discrete so that in both E³ and corresponding momentum space integration reduces to a summation. This would be achieved if discretisation is by lattice or QC so that one would obtain the desired constraint on discretizations. Thus Riemann hypothesis has excellent mathematical motivations to be true in TGD Universe! Freeman Dyson has represented a highly interesting speculation related to Riemann hypothesis and 1-dimensional quasicrystals (QCs). He discusses QCs and Riemann hypothesis briefly in his Einstein lecture.

   Dyson begins from the defining property of QC as discrete set of points of Euclidian space for which the spectrum of wave vectors associated with the Fourier transform is also discrete. What this says is that quasicrystal as also ordinary crystal creates discrete diffraction spectrum. This presumably holds true also in higher dimensions than D=1 although Dyson considers mostly D=1 case. Thus QC and its dual would correspond to discrete points sets. I will consider the consequences in TGD framework below.

   Dyson considers first QCs at general level. Dyson claims that QCs are possible only in dimensions D=1,2,3. I do not know whether this is really the case. In dimension D=3 the known QCs have icosahedral symmetry and there are only very few of them. In 2-D case (Penrose tilings) there is n-fold symmetry, roughly one kind of QC associated with any regular polygon. Penrose tilings correspond to n=5. In 1-D case there is no point group (subgroup of rotation group) and this explains why the number of QCs is infinite. For instance, so called PV numbers identified as algebraic integers, which are roots of any polynomial with integer coefficients such that all other roots have modulus smaller than unity. 1-D QCs is at least as rich a structure as PV numbers and probably much richer.

   Dyson suggests that Riemann hypothesis and its generalisations might be proved by studying 1-D quasi-crystals.

   1. If Riemann Hypothesis is true, the spectrum for the Fourier transform of the distribution of zeros of Riemann zeta is discrete. The calculations of Andrew Odlycko indeed demonstrate this numerically, which is of course not a proof. From Dyson's explanation I understand that it consists of sums of integer multiples nlog(p) of logarithms of primes meaning that the non-vanishing Fourier components are apart from overall delta function (number of zeros) proportional to

      F(n)= ∑_sk n^-is_k=ζ_D(is_k) , s_k=1/2+iy_k ,

      where s_k are zeros of Zeta. ζ_D could be called the dual of zeta with summation over integers replaced with summation over zeros. For other "energies" than E=log(n) the Fourier transform would vanish. One can say that the zeros of Riemann Zeta and primes (or p-adic "energy" spectrum) are dual. Dyson conjectures that each generalized zeta function (or rather, L-function) corresponds to one particular 1-D QC and that Riemann zeta corresponds to one very special 1-D QC.

   There are also intriguing connections with TGD, which inspire quaternionic generalization of Riemann Zeta and Riemann hypothesis.
   1. What is interesting that the same "energy" spectrum (logarithms of positive integers) appears in an arithmetic quantum field theory assignable to what I call infinite primes. An infinite hierarchy of second quantizations of ordinary arithmetic QFT is involved. A the lowest level the Fourier transform of the spectrum of the arithmetic QFT would consist of zeros of zeta rotated by π/2! The algebraic extensions of rationals and the algebraic integers associated with them define an infinite series of infinite primes and also generalized zeta functions obtained by the generalization of the sum formula. This would suggest a very deep connection with zeta functions, quantum physics, and quasicrystals. These zeta functions could correspond to 1-D QCs.
   2. The definition of p-adic manifold (in TGD framework) forces a discretisation of M⁴× CP₂ having interpretation in terms of finite measurement resolution. This discretization induces also dicretization of space-time surfaces by induction of manifold structure. The discretisation of M⁴ (or E³) is achieved by crystal lattices, by QCs, and perhaps also by more general discrete structures. Could lattices and QCs be forced by the condition that the lattice like structures defines a discrete distributions with discrete spectrum? But why this?
   3. There is also another problem. Integration is a problematic notion in p-adic context and it has turned out that discretization is unavoidable and also natural in finite measurement resolution. The inverse of the Fourier transform however involves integration unless the spectrum of the Fourier transform is discrete so that in both E³ and corresponding momentum space integration reduces to a summation. This would be achieved if discretisation is by lattice or QC so that one would obtain the desired constraint on discretizations. Thus Riemann hypothesis has excellent mathematical motivations to be true in TGD Universe!
   4. What could be the counterpart of Riemann Zeta in the quaternionic case? Quaternionic analog of Zeta suggests itself: formally one can define quaternionic zeta using the same formula as for Riemann zeta.
      1. Rieman zeta characterizes ordinary integers and s is in this case complex number, extension of reals by adding a imaginary unit. A naive generalization would be that quaternionic zeta characterizes Gaussian integers so that s in the sum ζ(s)=∑ n^-s should be replaced with quaternion and n by Gaussian integer. In octonionic zeta s should be replaced with octonion and n with a quaternionic integer. The sum is well-defined despite the non-commutativity of quaternions (non-associativity of octonions) if the powers n^-s are well-defined. Also the analytic continuation to entire quaternion/octonion plane should make sense and could be performed in a step wise manner by starting from real axis for s, extended to complex plane and then to quaternionic plane.
      2. Could the zeros s_k of quaternionic zeta ζ_H(s) reside at the 3-D hyper-plane Re(q)=1/2, where Re(q) corresponds to E⁴ time coordinate (one must also be able to continue to M⁴)? Could the duals of zeros in turn correspond to logarithms ilog(n), n Gaussian integer. The Fourier transform of the 3-D distribution defined by the zeros would in turn be proportional to the dual of ζ_D,H(is_k) of ζ_H. Same applies to the octonionic zeta.
      3. The assumption that n is ordinary integer in ζ_H would trivialize the situation. One obtains the distribution of zeros of ordinary Riemann zeta at each line s= 1/2+ yI, I any quaternionic unit and the loci of zeros would correspond to entire 2-spheres. The Fourier spectrum would not be discrete since only the magnitudes of the magnitudes of the quaternionic imaginary parts of "momenta" would be imaginary parts of zeros of Riemann zeta but the direction of momentum would be free. One would not avoid integration in the definition of inverse Fourier transform although the integrand would be constant in angular degrees of freedom.

   For background see the chapter Riemann Hypothesis and Physics.