Are there statistical observations about prime numbers showing that primes are not random? For example obviously primes are $1$ or $-1$ mod $6$, but are these remainder distributed equally? What I mean is, that if you derive some contraints on prime statements, do primes seem statistically random within the number set satisfying these constraints? Or are there some observations which are seen, but cannot be explained. I'm basically wondering if there is evidence that the "construction" of primes has some inherent deep complexity, that statements about primes are always limited and at some point only randomness can be assumed. I could compare that to fractal images where the generating equation is very simple, but the resulting images seem structured but still unpredictable. Hope I made my question clear :)

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en.wikipedia.org/wiki/Ulam_spiral –  Thomas Rot Mar 26 '12 at 11:22
The most prominent thing about primes that can be seen, but not explained might be the fact the zeros of the $\zeta$ function have $\Re(s)=1/2$. –  draks ... Mar 26 '12 at 11:23
The PNT for arithmetic progressions en.wikipedia.org/wiki/… gives that the primes +/-1 mod 6 question are asymptotically "distributed equally" in some sense. –  Zander Mar 26 '12 at 11:54
The Ulam Spiral would be a good example. However, I vaguely remember that this pattern has been explaned due to all prime numbers being of a particular form? –  Gerenuk Mar 26 '12 at 12:01
How about Chebyshev bias? Although primes equal to 1 and 3 are equally distributed mod 4 in some sense, the absolute bias also seems to exists. You may refer to this page for the details. –  sos440 Mar 26 '12 at 12:45

There is a heuristic probabilistic model of the primes due to Cramér based on the prime number theorem wherein a number $n$ has probability $\frac{1}{\log n}$ of being a prime. The idea is that statistical properties of these random numbers should with probability $1$ coincide with statistical properties of the primes themselves. So the question can be interpreted as follows:

Are there any results about the primes that disagree with the predictions of Cramér's model?

Of course Cramér's model predicts that about half of all prime numbers are even, which is silly, but this isn't hard to fix. More seriously, in 1985 Maier proved a result about the distribution of primes that disagrees with Cramér's model in a stronger way: details can be found in Granville's Harald Cramér and the distribution of primes.

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Clear as mud.

The primes aren't random. They are defined by a completely deterministic process.

Nevertheless, there are many situations in which a probabilistic model for primes makes predictions which agree with all the numerical evidence.

There are many observations about primes that cannot be explained, but I don't know what this has to do with randomness; there are just a lot of hard questions around.

If you could ask a precise question, maybe I could give a precise answer.

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The Ulam Spiral in the comments is a good answer. I think you are just not getting the question. –  Gerenuk Mar 26 '12 at 12:12
You are absolutely correct, I do not get the question. You have to decide whether that is my fault, or the fault of the question. What does it mean, "observations which are seen, but cannot be explained"? What does it mean, "statements about primes are always limited and at some point only randomness can be assumed"? I did the best I could to understand this word-salad. Evidently, I failed. Better luck next time. –  Gerry Myerson Mar 26 '12 at 12:21
It's OK. Others are more successful answering. I check out the article on Ulam. –  Gerenuk Mar 26 '12 at 12:23