Bounds
on S(T,x) imply bounds on moments

•__Folklore theorem__ (known as early as the 60’s by Chandrasekharan, Narasimhan, Selberg):

–If S(T,x) = Oe(Tb+e) for some ½ · b < 1, then

s-TT |L(½ + it)|2 dt = Oe(T1 + e + (2b-1) d),

s-TT |L(½ + it)|2 dt = Oe(T1 + e + (2b-1) d),

•Where d = the degree of the L-function

•E.g. L-function comes from GL(d**,**AQ).

•Thus b = 1/2
is very hard to achieve because it gives the optimal bound Oe(T1+e) .

•GL(3) result of Oe(T3/4+e) unfortunately does not give new moment information.

•Voronoi-style summation formulas with Schmid give an implication between:

–squareroot cancellation in sums of *d-1*-hyperkloosterman sums weighted by an, and

–Optimal cancellation S(T,x) = Oe(T1/2+e) – and therefore Lindelöf also.