Ratios June 4, 2009
To get the ratios conjecture • Follow the recipe for moments • Replace the numerator L’s by apprx fnc eq • Replace the denominator L’s by their full Dirichlet series • Multiply out • Bring the average over the family inside • Replace averages by their expected values using the appropriate harmonics of the family • Extend all coefficient sums, extract zeta’s
Montgomery’s Pair Correlation Conjecture
Picture by A. Odlyzko 79 million zeros around the th zero
First 100000 zeros
zeros around the th zero
Refined pair-correlation conjecture (Bogomolny-Keating, Conrey-Snaith)
We want to evaluate T 1-a 1/2 a
Hejhal, 1994 - triple correlation where the Fourier transform of f has support on the hexagon with vertices (1,0),(0,1),(-1,1),(-1,0),(0,-1),(1,-1), and
Rudnick and Sarnak, 1996 n-correlation: Bogomolny and Keating, 1995,1996 Heuristic using Hardy-Littlewood conjecture to obtain large T scaling limit Scaling limit for the n-point correlation function, again with restricted support of the Fourier transform of the test function.
A,B,Q,P are expressions involving primes (see Bogomolny, Keating, Phys.Rev.Lett.,1996)
Steve Gonek proved this, assuming RH, for k=1. The RMT analogue of the conjecture is a theorem due to Chris Hughes.