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In the Princeton Companion of Mathematics, the Analytic Number Theory section, the author mentions what he calls Gauss-Kramer model, which is simply modeling the integers on a countable sequence of random variables $\{ X_{n} \}$, where the variable $X_{n}$ has probability of $1/log(n)$ of being 1 and otherwise 0. Each variable $X_{n}$ represents whether the number n is prime or not. The Gauss-Kramer model conjectures that statements about the distribution of primes correspond to the statements on the countable collection of random variables with probability 1 of being true.

Question: Is anyone aware of a good exposition of the Gauss-Kramer model and any attempt to make it more rigorous using Model Theory or through other routes?

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    $\begingroup$ Usually written Cramer. I remember this MO question $$ $$ mathoverflow.net/questions/36897/… $$ $$ I will look for others. To answer your second question, it is known to be faulty by 1985 work of Meier. $\endgroup$
    – Will Jagy
    Commented Jun 7, 2011 at 3:24
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    $\begingroup$ Alright, Maier. see $$ $$ mathoverflow.net/questions/11978/… $$ $$ $\endgroup$
    – Will Jagy
    Commented Jun 7, 2011 at 3:25
  • $\begingroup$ Thanks a lot for the suggestion, I am reading the synopsis of Meier's work currently. $\endgroup$
    – Mohamed Alaa El Behairy
    Commented Jun 7, 2011 at 16:28

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Chapter 3 of The Prime Numbers and Their Distribution by Gérald Tenenbaum and Michel Mendès France has a nice exposition of the model, including modifications indicated by Maier's discovery.

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  • $\begingroup$ Thanks a lot for your suggestion. Has there been any attempt to make the model more rigorous? $\endgroup$
    – Mohamed Alaa El Behairy
    Commented Jun 7, 2011 at 16:29
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    $\begingroup$ I don't think there has been much hope of making it rigorous, because the conclusions it yields are so far beyond what can be proved today. For example, if $p_k$ denotes the k-th prime, it is known that $p_{k+1} = p_k + O(p_k^{\alpha + \varepsilon})$ with $\alpha = 0.525$. The Riemann Hypothesis would allow $\alpha = 0.5$. But the Cramér model would yield $p_{k+1} = p_k + O(\log^2(p_k))$. $\endgroup$
    – Marius Overholt
    Commented Jun 7, 2011 at 16:48
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You might want to look at:

1) "Harold Cramér and the distribution of prime numbers" by Andrew Granville http://www.dms.umontreal.ca/~andrew/PDF/cramer.pdf

2) "The distribution of prime numbers" by K. Soundararajan http://arxiv.org/pdf/math/0606408v1

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