Further progress in the number theoretic anatomy of zeros of zeta

I wrote for some time ago an article about number theoretical universality (NTU) in TGD framework and ended up with the conjecture that the non-trivial zeros of zeta can be divided into classes C(p) labelled by primes p such that for zeros y in given class C(p) the phases p^iy are roots of unity. The impulse leading to the idea came from an argument of Dyson referring to the evidence that the Fourier transform for the locus of non-trivial zeros of zeta is a distribution concentrated on powers of primes.

There is a very interesting blog post by Mumford , which led to a much more precise formulation of the idea and improved view about the Fourier transform hypothesis: the Fourier transform must be defined for all zeros, not only the non-trivial ones and trivial zeros give a background term allowing to understand better the properties of the Fourier transform.

Mumford essentially begins from Riemann's "explicit formula" in von Mangoldt's form.

∑_p∑_{n≥ 1} log(p)δ_pⁿ(x)= 1- ∑_k x^s_k-1 - 1/x(x²-1) ,

where p denotes prime and s_k a non-trivial zero of zeta. The left hand side represents the distribution associated with powers of primes. The right hand side contains sum over cosines

∑_k x^s_k-1= 2∑_k cos(log(x)y_k)/x^1/2 ,

where y_k ithe imaginary part of non-trivial zero . Apart from the factor x^-1/2 this is the just Fourier transform over the distribution of zeros.

There is also a slowly varying term 1-1/x(x²-1), which has interpretation as the analog of the Fourier transform term but sum over trivial zeros of zeta at s=-2n, n>0. The entire expression is analogous to a "Fourier transform" over the distribution of all zeros.

Therefore the distribution for powers of primes is expressible as "Fourier transform" over the distribution of both trivial and non-trivial zeros rather than only non-trivial zeros as suggested by numerical data to which Dyson referred to). Trivial zeros give a slowly varying background term large for small values of argument x (poles at x=0 and x=1 - note that also p=0 and p=1 appear effectively as primes) so that the peaks of the distribution are higher for small primes.

The question was how can one obtain this kind of delta function distribution concentrated on powers of primes from a sum over terms cos(log(x)y_k) appearing in the Fourier transform of the distribution of zeros. Consider x=pⁿ. One must get a constructive interference. Stationary phase approximation is in terms of which physicist thinks. The argument was that a destructive interference occurs for given x=pⁿ for those zeros for which cosine does not correspond to a real part of root of unity as one sums over such y_k: random phase approximation gives more or less zero. To get something nontrivial y_k must be proportional to 2π× n(y_k)/log(p) in class C(p) to which y_k belongs. If the number of these y_k:s in C(p) is infinite, one obtains delta function in good approximation by destructive interference for other values of argument x.

The guess that the number of zeros in C(p) is infinite is encouraged by the behaviors of the densities of primes one hand and zeros of zeta on the other hand. The number of primes smaller than real number x goes like

π(x)=#(primes <x)∼ x/log(x)

in the sense of distribution. The number of zeros along critical line goes like

#(zeros <t) = (t/2π) × log(t/2π)

in the same sense. If the real axis and critical line have same metric measure then one can say that the number of zeros in interval T per number of primes in interval T behaves roughly like

#(zeros<T)/#(primes<T) =log(T/2π)×log(T)/2π

so that at the limit of T→ ∞ the number of zeros associated with given prime is infinite. This asumption of course makes the argument a poor man's argument only.

See the chapter Unified Number Theoretic Vision of "Physics 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.