•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 anequal to the
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¸1ann-s, which beyond GL(2) is thought to be a problem as difficult as the Lindelof conjecture.