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 posterior I
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.
A posterior II
Thank you for all of many the comprehensive answers and discussions. This is a quite classic question on MSE and meanwhile we moved much forward. You are right that primes are not random as per above question. Indeed we could show that they are in their sequence some type of "deterministic chaos". We don't need the Riemann function for this purpose. The primes sequence is a so called "ordered iterative sequence". Meanwhile this has been further elaborated by this source: "The Secret Harmony of Primes" (ISBN 978-9176370001) http://a.co/iIHQqR8 Some of you correctly referred to sieving. It is crucial however that we regard sieving procedures as a subset of "interference" (incl. frequencies and amplitudes). We can iteratively apply interference rules in order to gain from the first prime 2, the next ordered sequence. This can be iterative continued in an "ordered" way and within exact boundaries of p-squares (for 100% certainty). Indeed, in order to construct an ordered sequence of primes you just need to begin with 2. The Riemann approach is charming but would raise difficulties since we don't have yet a proof of the hypothesis that connects the order of the non-trivial zeros with the primes. So if you apply Riemann, as some colleagues here suggest, we would need to say at any time in the begin of your argumentation something like "provided the Riemann hypothesis would be true...". Having in mind that the very unique rule that primes follow, is that in an interference scheme all odd prime frequencies dance on the base frequency of 2 (ordered iterative sequence), one may even give it a thought to something of a parallel in the Riemann transformed world, that all non-trivial zeros dance on 1/2. But latter remains not more than a tempting trivial speculation yet.
Terence Tao wrote about it, I've found this video and there's also one article called: Structure and randomness in the prime numbers, I've read it in the book: An Invitation to Mathematics: From Competitions to Research, by Dierk Schleicher and Malte Lackmann.
The article I mentioned can be found here.