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.
Best Answer
The primes are not randomly distributed. They are completely deterministic in the sense that the $n$th prime can be found via sieving. We speak loosely of the probability that a given number $n$ is prime $({\bf P}(n\in {\mathbb P}) \approx 1/\log n)$ based on the prime number theorem but this does not change matters and is largely a convenience.
Some confusion is maybe due to the use of probabilistic methods to prove interesting things about primes and because once we put the sieve aside the primes are pretty inscrutable. They seem random in the sense that we cannot predict their appearance in some formulaic way.
On the other hand the primes have properties associated more or less directly with random numbers. It has been shown that the form of the "explicit formulas" (such as that of von Mangoldt) obeyed by zeros of the $\zeta$ function imply what is known as the GUE hypothesis: roughly speaking the zeros of the $\zeta$ function are spaced in a non-random way. The eigenvalues of certain types of random matrices share this property with the zeros. There is a proof of this.$^1$
So it can be said that the primes are a deterministic sequence that via the $\zeta$ function share a salient feature with putatively random sequences.
In response to the particular question, "random" here is the "random" of random matrix theory. The paper trail is pretty clear from the work below and it's not a subject that fits into an answer box.
$^1$ Rudnick and Sarnak, Zeros of Principal L-Functions and Random Matrix Theory, Duke Math. J., vol. 81 no. 2 (1996).