Higher Rank?

•Only general result is
the trivial bound S(T,x) = O(T).

•Theorem (Miller, 2004)
For cusp forms on GL(3,**Z**)nGL(3,**R**) and an equal to the
standard

L-function coefficients, S(T,x) = Oe(T3/4+e).

L-function coefficients, S(T,x) = Oe(T3/4+e).

•This is halfway
between the trivial O(T) and optimal Oe(T1/2+) bounds.

•We will see that the
full conjecture implies the correct
order of magnitude for the second moment of L(s)=ån¸ 1an n-s, which beyond GL(2) is thought to be a problem as difficult as the Lindelof conjecture.