A Study of the Riemann Zeta Zeros

The goals of the proposed research are: 1) To obtain additional concrete computational evidence §2.1 of the unknown structure on the imaginary parts of the non-trivial zeros of the Riemann zeta function, called herein the Riemann spectrum, following the method from [1]; specifically, the PI will compute correlations between histograms of random variables Xp (§2); 2) To prove the that the density distributions from [1] follows from the Landau’s formula for the Riemann frequencies [5], p.80, and to prove its equivalence with the RiemannMangoldt-Weil-Delsarte exact formula in terms of distributions, as presented herein, following [26] (§3.2, §3.4); 3) To search for a theoretical interpretation of the results in [1, 26], in terms of adelic and idelic characters (§4); 4) To investigate if the partial order << on the prime numbers [2] is reflected via duality in the Riemann spectrum, by a direct study of the convergence of the distributional Fourier transform along Pratt trees (q << p), instead of the convergence of the usual partial sums corresponding to the size of primes (§1.2.1); 5) Explore the relations between the Riemann spectrum, adelic characters and distributions, in terms of Hecke (idelic) characters, local zeta integrals (e.g. Mellin transform) and ω-eigendistributions from [4], as a preparation for the next research phase (§5). The computational investigations will be carried out using the public CAS SAGE [36].