Ratios
June 4, 2009
Euler Product
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
Application to Mollifying
Use Perron’s formula
RATIOS THEOREM (UNITARY)
RATIOS THEOREM (ORTHOGONAL)
RATIOS THEOREM (SYMPLECTIC)
Ratios conjecture (zeta)
Ratios conjecture (zeta)
Application to pair correlation
Montgomery, 1971 – pair correlation
Montgomery’s Pair Correlation
Conjecture
Picture by
A. Odlyzko
79 million zeros
around the
th zero
First
100000
zeros
zeros
around the
th zero
Bogomolny and Keating
Refined pair-correlation conjecture
(Bogomolny-Keating, Conrey-Snaith)
The ratios approach to lower order terms
We want to evaluate
T
1-a
1/2 a
Move contours to the right, becomes
Assuming the ratios conjecture:
with
Difference between theory and
numerics:
Assuming the ratios conjecture:
with
For large T:
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
n-correlation:
Bogomolny and Keating, 1995,1996
Heuristic using Hardy-Littlewood conjecture to
obtain large T scaling limit
Rudnick and Sarnak, 1996
Scaling limit for the n-point correlation function,
again with restricted support of the Fourier
transform of the test function.
Triple correlation using ratios:
A,B,Q,P are expressions involving primes
(see Bogomolny, Keating, Phys.Rev.Lett.,1996)
Applications to lower order
terms in one-level densities
One-level density
Application to discrete
moments
Steve Gonek proved this, assuming RH, for k=1. The RMT
analogue of the conjecture is a theorem due to Chris Hughes.
Lower order terms when k=2
The fourth moment
It would be nice to numerically check this formula, with all of the
terms included.
The End