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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. –  Vÿska 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: arxiv.org/abs/1102.3648 –  al-Hwarizmi Jun 16 '13 at 9:55
    
Oh! Thanks for the reference. –  Vÿska 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 math.stackexchange.com/questions/680122/…. –  Conifold Sep 22 at 20:57

2 Answers 2

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? –  Vÿska 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. –  Vÿska Jun 15 '13 at 22:18
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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 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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