There are so many results and conjectures having to do with the mean-values
or moments of L-functions in families that it is difficult to mention
all of them. So we will stick with some of the more familiar ones.
Moments of . Let
Then asymptotic formulae are known for and . A good estimation for seems to be far out of reach of today's technology (maybe we need to know
more about automorphic forms on GL3?) but would be a very important milestone. Conjectures based on number theory are known for . Conjectures based on random matrix theory are known for all real .
Moments of . Let be a real, primitive, quadratic character to the modulus (i.e. a Kronecker symbol).
Then asymptotics are known for . A conjecture based on number theory is known for . Conjectures based on random matrix theory are known for all real . The fourth moment is just slightly out of reach of current technology.
Moments of automorphic L-functions. Let be a normalized newform of weight and level 1 and let be the associated -function
(with critical strip .) Let
Then asymptotic formulae are known for
. Conjectures based on random matrix theory are known for all real
Is the fifth moment doable?
Quadratic twists of automorphic L-functions. Let be a fixed newform (for example the newform associated with a given elliptic curve).
Let be the associated L-function and the twist by the primitive quadratic character . It is possible to evaluate the
but can one do the second moment
Again, this is just at the edge of what can be done by today's techniques
and is a problem worthy of study.
Special families. Recently the bound
has been obtained, where denotes the set of newforms of
weight and level (no character), and is the real character to the modulus where is odd and squarefree. This estimate has been used to bound fourier coefficients of half-integral weight forms and (its analogue for Maass forms has been used) to bound
. Can one strengthen this estimate to give
an asymptotic formula (perhaps for restricted to prime values)? Are there other situations where the underlying arithmetic is so fortuitous so as to give such a strong bound?
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for L-functions and Random Matrix Theory.