Bounds
on S(T,x) for general cusp forms (on GL(d))

•Recall that we expect S(T,x) = ån6T an e(nx) to be Oe(T1/2+e)
when an are the
coefficients of an entire L-function.

–according to the Langlands/Selberg/Piatetski-Shapiro
philosophy, these are always
L-functions of cusp forms on GL(2,**A****Q**).

•__Main known result__: S(T,x) = Oe(T1/2+e).

for cusp forms on GL(2) (degree 2 L-functions)

for cusp forms on GL(2) (degree 2 L-functions)

–For holomorphic cusp forms, this is classical and straightforward to prove

–But for Maass forms this is much more subtle.

–Importance: used in Hardy-Littlewood’s seminal method to
prove z(s) has infinitely many zeroes on its critical line

(we will see this again later).

(we will see this again later).