# A note on the statistics of Riemann zeros

## TL;DR

This paper reviews evidence of algebraic and analytic structures in the Riemann zeros, exploring their distribution and duality with primes through statistical and group-theoretic methods.

## Contribution

It introduces a statistical approach to analyze the structure of Riemann zeros and primes, emphasizing the duality and symmetry properties involved.

## Key findings

- Identified resonances at generators of symmetry groups of finite fields.
- Computed correlation coefficients indicating structured relationships among zeros.
- Proposed a program for further algebraic analysis 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.