Framework for primes vs random matrices - MathOverflow most recent 30 from http://mathoverflow.net 2013-05-22T09:17:00Z http://mathoverflow.net/feeds/question/65633 http://www.creativecommons.org/licenses/by-nc/2.5/rdf http://mathoverflow.net/questions/65633/framework-for-primes-vs-random-matrices Framework for primes vs random matrices Neil Strickland 2011-05-21T08:37:20Z 2011-05-23T09:55:32Z <p>This is inspired by <a href="http://mathoverflow.net/questions/65584" rel="nofollow">http://mathoverflow.net/questions/65584</a> , but I think it is sufficiently different to ask separately. </p> <p>We can formalise probability in terms of an algebra $A$ of random variables equipped with an expectation function $E:A\to\mathbb{R}$ subject to certain axioms. For example, we can consider the algebra $A(n)$ of conjugation-invariant functions on $U(n)$, equipped with the Haar integral. We could then take some kind of limit $A(\infty)$. I don't know exactly what the details of the limiting process should be. Alternatively, we can take $B(a,b,c)$ to be some algebra of functions of arithmetical interest on the set $\{a,a+1,\dotsc,a+b\}^c$, where we think of $b$ as being quite large, and $a$ as being much larger. If we choose these algebras the right way, then it should be possible to compare $B(a,b,c)$ with $B(a',b',c')$ for suitable values of $a'$, $b'$ and $c'$. The things that people say about primes and random matrices suggest to me that it should be possible to construct some kind of limiting object $B(\infty)$, and to conjecture that $A(\infty)$ is isomorphic to $B(\infty)$ as algebras-with-expectation; and hopefully this would imply all the other conjectures that people talk about. Perhaps $B(\infty)$ could be described in terms of adeles.</p> <p>Has anything like this been done? Or is it known to be impossible? </p>