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

1/2

T

a1-a

Move contours to the right, becomes

with

Assuming the ratios conjecture:

Difference between theory and numerics:

with

Assuming the ratios conjecture:

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

Rudnick and Sarnak, 1996 Scaling limit for the n-point correlation function,

again with restricted support of the Fourier

transform of the test function.

n-correlation:

Bogomolny and Keating, 1995,1996

Heuristic using Hardy-Littlewood conjecture to

obtain large T scaling limit

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