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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

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.

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I've seen you edited it, does it make sense to talk about noise in the prime numbers? I'm guessing it's weird but it's only a gut feeling. Still waiting for the more qualified person. – Voyska Jun 16 '13 at 9:53
In fact we talk about noise in primes. There is quite extensive literature on this, see for instance here: – al-Hwarizmi Jun 16 '13 at 9:55
Oh! Thanks for the reference. – Voyska Jun 17 '13 at 1:41
Another connection is to Cramer's probabilistic model, a conjecture that although primes are not random in some specified ways they behave as if they were random…. – Conifold Sep 22 '14 at 20:57

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.

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this is interesting general information but hardly focus on a clear and cristalized answer to the question: primes randomly distributed? so then what is 'random' in this context? – al-Hwarizmi Jun 15 '13 at 22:09
Did you read the article? – Voyska Jun 15 '13 at 22:12
yes, I did. I also know the vids of Tery well. The book is new to me. – al-Hwarizmi Jun 15 '13 at 22:16
Also, I believe that there isn't a random in this context - I guess random means the absence of pattern, even in this case. But I'm not sure, I hope someone more qualified answers that for you. Good luck. – Voyska Jun 15 '13 at 22:18
Section 4 of the article seems to very clearly cover your question (and could perhaps be added to this answer) : the primes behave very much like a random set with density approximately their density would, with the specific 'sieve constraints' (mod smaller primes) modifying constants on the random behavior (see e.g. the twin prime constant) but generally not changing asymptotics at all. – Steven Stadnicki Jun 17 '13 at 20:13

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 $(p~(n\in P) \approx 1/\log n)$ 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).

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The simple answer is no they are not random. Though I can not give you the mathematical formula to prove this, I can share with you the title of a book someone just suggested I read by Mark Kac, called Statistical Independence in Probability. I can also point out that since Prime numbers are factual things that are always going to be in the same numerical location no matter what number system you use, that they can not be random (random in the simplest layman's understanding of that word that is) they must therefor have a pattern. We just do not yet "FULLY" comprehend it.

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Good points. Can you expand on "since Prime numbers are factual things that are always going to be in the same numerical location no matter what number system you use, that they can not be random" please, Is there any related theorem or conjuncture or result related to this statement? – Arjang Jul 31 '15 at 7:21
up vote 0 down vote accepted

The answer to this question is now given by Eupraxis1981: I totally agree with the others who have commented, in that no arbitrary sequence of numbers is inherently random. For example, the sequence of primes (as you've pointed out) have been called random, yet we know that they can be represented by a (admittedly complex) system of Diophantine equations. So they form a rather well-ordered group, but just incredibly complicated... >>>here

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The prime numbers is the set of numbers derived by removing the set of composite numbers from the set of natural numbers.

The pattern that the natural numbers follow is easily understood. The set of composite numbers also follow a discernable pattern:it is the union of the factors of 2, the factors of 3 etc.

Therefore the prime numbers as defined in the first paragraph must follow a pattern. No doubt the pattern is complex and we are yet to discover a way of defining it using a formula or formulas.

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