I have made several number theoretic peculations related to the possible role of zeros of Riemann zeta in coupling constant evolution. The basic problem is that it is not even known whether the zeros of zeta are rationals, algebraic numbers or genuine transcendentals or belong to all these categories. Also the question whether number theoretic analogs of ζ defined for p-adic number fields could make sense in some sense is interesting.
1. Is number theoretic analog of ζ possible using Log(p) instead of log(p)?
The definition of Log(n) based on factorization Log(n)==∑pkpLog(p) allows to define the number theoretic version of Riemann Zeta ζ(s)=∑ n-s via the replacement n-s=exp(-log(n)s)→ exp(-Log(n)s).
Remark: One could modify also the definition of ζ for complex numbers by replacing exp(log(n)s) with exp(Log(n)s) with Log(n)= ∑p kpLog(p) to get the prime factorization formula. I will refer to this variant of zeta as modified zeta below.
- In suitable region of plane number-theoretic Zeta would have the usual decomposition to factors via the replacement 1/(1-p-s)→ 1/(1-exp(-Log(p)s). p-Adically this makes sense for s= O(p) and thus only for a finite number of primes p for positive integer valued s: one obtains kind of cut-off zeta. Number theoretic zeta would be sensitive only to a finite number of prime factors of integer n.
- This might relate to the strong physical indications that only a finite number of cognitive representations characterized by p-adic primes are present in given quantum state: the ramified primes for the extension are excellent candidates for these p-adic primes. The size scale n of CD could also have decomposition to a product of powers of ramified primes. The finiteness of cognition conforms with the cutoff: for given CD size n and extension of rationals the p-adic primes labelling cognitive representations would be fixed.
- One can expand the regions of converge to larger p-adic norms by introducing an extension of p-adics containing e and some of its roots (ep is automatically a p-adic number). By introducing roots of unity, one can define the phase factor exp(-iLog(n)Im(s)) for suitable values of Im(s). Clearly, exp(-ipIm(s))/π(p)) must be in the extension used for all primes p involved. One must therefore introduce prime roots exp(i/π(p)) for primes appearing in cutoff. To define the number theoretic zeta for all p-adic integer values of Re(s) and all integer values of Im(s), one should allow all roots of unity (ep(i2π/n)) and all roots e1/n: this requires infinite-dimensional extension.
- One can thus define a hierarchy of cutoffs of zeta: for this the factorization of Zeta to a finite number of "prime factors" takes place in genuine sense, and the points Im(s)= ikπ(p) give rise to poles of the cutoff zeta as poles of prime factors. Cutoff zeta converges to zero for Re(s)→ ∞ and exists along angles corresponding to allowed roots of unity. Cutoff zeta diverges for (Re(s)=0, Im(s)= ik π(p)) for the primes p appearing in it.
2. Could the values of 1/αK be given as zeros of ζ or of modified ζ
I have discussed the possibility that the zeros s=1/2+iy of Riemann zeta at critical line correspond to the values of complex valued Kähler coupling strength αK: s=i/αK (see this). The assumption that piy is root of unity for some combinations of p and y [log(p)y =(r/s)2π] was made. This does not allow s to be complex rational. If the exponent of Kähler action disappears from the scattering amplitudes as M8-H duality requires, one could assume that s has rational values but also algebraic values are allowed.
3. The argument of Dyson for the Fourier transform of the characteristic function for the set of zeros of ζ
- If one combines the proposed idea about the Log-arithmic dependence of the coupling constants on the size of CD and algebraic extension with s=i/αK hypothesis, one cannot avoid the conjecture that the zeros of zeta are complex rationals. It is not known whether this is the case or not. The rationality would not have any strong implications for number theory but the existence irrational roots would have (see this). Interestingly, the rationality of the roots would have very powerful physical implications if TGD inspired number theoretical conjectures are accepted.
The argument discussed below however shows that complex rational roots of zeta are not favored by the observations about the Fourier transform for the characteristic function for the zeros of zeta. Rather, the findings suggest that the imaginary parts (see this) should be rational multiples of 2π, which does not conform with the vision that 1/αK is algebraic number. The replacement of log(p) with Log(p) and of 2π with is natural p-adic approximation in an extension allowing roots of unity however allows 1/αK to be an algebraic number. Could the spectrum of 1/αK correspond to the roots of ζ or of modified ζ?
- A further conjecture discussed was that there is 1-1 correspondence between primes p≈ 2k, k prime, and zeros of zeta so that there would be an order preserving map k→ sk. The support for the conjecture was the predicted rather reasonable coupling constant evolution for αK. Primes near powers of 2 could be physically special because Log(n) decomposes to sum of Log(p):s and would increase dramatically at n=2k slightly above them.
In an attempt to understand why just prime values of k are physically special, I have proposed that k-adic length scales correspond to the size scales of wormhole contacts whereas particle space-time sheets would correspond to p≈ 2k. Could the logarithmic relation between Lp and Lk correspond to logarithmic relation between p and π(p) in case that π(p) is prime and could this condition select the preferred p-adic primes p?
Consider now the argument suggesting that the roots of zeta cannot be complex rationals. On basis of numerical evidence Dyson (see this) has conjectured that the Fourier transform for the characteristic function for the critical zeros of zeta consists of multiples of logarithms log(p) of primes so that one could regard zeros as one-dimensional quasi-crystal.
This hypothesis makes sense if the zeros of zeta decompose into disjoint sets such that each set corresponds to its own prime (and its powers) and one has piy= Um/n=exp(i2π m/n) (see the appendix of this). This hypothesis is also motivated by number theoretical universality (see this).
Remark: This picture allows many variants. For instance, if one assumes standard zeta, one could consider the possibility that the roots yp associated with p and giving rise to constructive interference are of form y= q×(Log(p)/log(p))× sin(2π/p)p, q=r/s.
- One can re-write the discrete Fourier transform over zeros of ζ at critical line as
f(x)= ∑y exp(ixy)) , y=Im(s) .
The alternative form reads as
f(u) =∑s uiy , u=exp(x) .
f(u) is located at powers pn of primes defining ideals in the set of integers.
For y=pn one would have piny=exp(inlog(p)y). Note that k=nlog(p) is analogous to a wave vector. If exp(inlog(p)y) is root of unity as proposed earlier for some combinations of p and y, the Fourier transform becomes a sum over roots of unity for these combinations: this could make possible constructive interference for the roots of unity, which are same or at least have the same sign. For given p there should be several values of y(p) with nearly the same value of exp(inlog(p)y(p)) whereas other values of y would interfere deconstructively.
For general values y= xn x≠ p the sum would not be over roots of unity and constructive interference is not expected. Therefore the peaking at powers of p could take place. This picture does not support the hypothesis that zeros of zeta are complex rational numbers so that the values of 1/αK correspond to zeros of zeta and would be therefore complex rationals as the simplest view about coupling constant evolution would suggest.
- What if one replaces log(p) with Log(p) =p/π(p), which is rational and thus ζ with modified ζ? For large enough values of p Log(p)≈ log(p) finite computational accuracy does not allow distinguish Log(p) from log(p). For Log(p) one could thus understand the finding in terms of constructive interference for the roots of unity if the roots of zeta are of form s= 1/2+i(m/n)2π. The value of y cannot be rational number and 1/αK would have real part equal to y proportional to 2π which would require infinite-D extension of rationals. In p-adic sectors infinite-D extension does not conform with the finiteness of cognition.
- Numerical calculations have however finite accuracy, and allow also the possibility that y is algebraic number approximating rational multiple of 2π in some natural manner. In p-adic sectors would obtain the spectrum of y and 1/αK as algebraic numbers by replacing 2π in the formula is= αK= i/2+ q× 2π, q=r/s, with its approximate value:
2π→ sin(2π/n)n= in/2(exp(i2π/n)- exp(-i2π/n))
for an extension of rationals containing n:th of unity. Maximum value of n would give the best approximation. This approximation performed by fundamental physics should appear in the number theoretic scattering amplitudes in the expressions for 1/αK to make it algebraic number.
y can be approximated in the same manner in p-adic sectors and a natural guess is that
n=p defines the maximal root of unity as exp(i2π/p). The phase exp(ilog(p)y) for y= q sin(2π/n(y)), q=r/s, is replaced with the approximation induced by log(p)→ Log(p) and 2π→ sin(2π/n)n giving
exp(ilog(p)y) → exp(iq(y) sin(2π/n(y))p/π(p)) .
If s in q=r/s does not contain higher powers of p, the exponent exists p-adically for this extension and can can be expanded in positive powers of p as
∑n inqn sin(2π/p)n (p/π(p))n .
This makes sense p-adically.
Also the actual complex roots of ζ could be algebraic numbers:
s= i/2+ q× sin(2π/n(y))n(y) .
If the proposed correlation between p-adic primes p≈ 2k, k prime and zeros of zeta predicting a reasonable coupling constant evolution for 1/αK is true, one can have naturally, n(y)=p(y), where p is the p-adic prime associated with y: the accuracy in angle measurement would increase with the size scale of CD. For given p there could be several roots y with same p(y) but different q(y) giving same phases or at least phases with same sign of real part.
Whether the roots of modified ζ are algebraic numbers and at critical line Re(s)=1/2 is an interesting question.
For background see the chapter The Recent View about Twistorialization in TGD Framework or the article with the same title.