The mean-value conjectures

For convenience we state examples of the three main conjectures for moments in families. There are no open questions in this article (except how to prove these conjectures!). For simplicity, we state the conjectures only for integral moments.

Unitary

Let

Conjecture.

where

Symplectic

Let be a real, primitive, quadratic character to the modulus (i.e. a Kronecker symbol) and let

where .

Conjecture.

where

Orthogonal

Let be a normalized newform of weight and level (where is prime) (we write and let be the associated -function (with critical strip .) Let

where

Conjecture.

where is a product over primes (which can be worked out for any , but for which we don't have a simple closed form expression);

and so on.

Back to the main index for L-functions and Random Matrix Theory.