The variance of the number of prime polynomials in short intervals and   in residue classes
  We resolve a function field version of two conjectures concerning the variance of the number of primes in short intervals (Goldston and Montgomery) and in arithmetic progressions (Hooley). A crucial ingredient in our work are recent equidistribution results of N. Katz. 
J. P. Keating and Z. Rudnick
math.NT