Arithmetic statistics in function fields

Wed, 23/12/2015 - 10:30
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Abstract: 
One of the most famous conjectures in number theory is the Hardy-Littlewood conjecture, which gives an asymptotic for the number of integers n up to X such that for a given tuple of integers a_1,.., a_k all the numbers n+a_1,.., n+a_k  are prime. This quantifies and generalises the twin-prime conjecture.

 

Function field analogue of this problem has recently been resolved in the limit of large finite field size q by Lior Bary-Soroker. However, in this limit the correlations disappear: the arithmetic functions become uncorrelated. It is therefore important to understand the terms of lower order in q, which must account for the correlations. We compute averages of these terms which detect correlations. Our results show that there is considerable cancellation in the averaging and have implications for the rate at which correlations disappear when q tends to infinity. This is a joint work with Jon Keating