A note on the statistics of Riemann zeros

Abstract

Evidence of an algebraic/analytic structure of the Riemann Spectrum, consisting of the imaginary parts of the corresponding zeros, is reviewed, with emphasis on the distribution of the image of the primes under the Cramer characters $X_p(t)=p^{it}$. The duality between primes and Riemann zeros, expressed traditionally as the Riemann-Mangoldt exact equation, is further used to investigate from a statistical point of view, the correspondence between the POSet structure of prime numbers and this yet unknown structure of R-Spec. Specifically, the statistical correlation coefficient $c(p,q)=<X_p,X_q>$ is computed, noting "resonances" at the generators $q$ of the symmetry group $Aut_{Ab}(F_p)$ of finite field $F_p$. A program for further studying the Riemann zeros from a pro-algebraic point of view, is presented.