There is a famous citation that says "It is evident that the primes are randomly distributed but, unfortunately, we don't know what 'random' means." R. C. Vaughan (February 1990)
I have this very clear but rather broad question that might be answered by different opinions and view points. However, my question is really not targeting an intuitive or philosophical answer, and I beg you for view points with a strength of mathematical foundation.
are primes randomly distributed? so then what is 'random' in this context?
A possible hint comes perhaps from the theory of complex dynamical systems.
It can be difficult to tell from data whether a physical or other observed process is random or chaotic, because in practice no time series consists of pure 'signal.' There will always be some form of corrupting noise, even if it is present as round-off or truncation error. Thus any real time series, even if mostly deterministic, will contain some randomness. All methods for distinguishing deterministic and stochastic processes rely on the fact that a deterministic system always evolves in the same way from a given starting point.(ref 1, 2, 3, and "Distinguishing random from chaotic data") - complying to latter, remind that every prime $p$ can be trivially identified by a sieving that applies prior primes $q<p$ so it is possible to determine that somehow the system evolves in the same way from a given starting point. Of course to take into account that time must be substituted by a walking index as well.