  The current research regarding the Riemann zeros suggests the existence of a
non-trivial algebraic/analytic structure on the set of Riemann zeros. The
duality between primes and Riemann zeta function zeros suggests some new goals
and aspects to be studied: {\em adelic duality} and the {\em POSet of prime
numbers}.
  The article presents computational evidence of the structure of the imaginary
parts $t$ of the non-trivial zeros of the Riemann zeta function $\rho=1/2+it$,
called in this article the {\em Riemann Spectrum}, using the study of their
distribution.
  The novelty represents in considering the associated characters $p^{it}$,
towards an algebraic point of view, than rather in the sense of Analytic Number
Theory. This structure is tentatively interpreted in terms of adelic
characters, and the duality of the rationals.
  Second, the POSet structure of prime numbers studied, is tentatively mirrored
via duality in the Riemann spectrum. A direct study of the convergence of their
Fourier series, along Pratt trees, is proposed.
  Further considerations, relating the Riemann Spectrum, adelic characters and
distributions, in terms of Hecke idelic characters, local zeta integrals
(Mellin transform) and $\omega$-eigen-distributions, are explored following.
