Bounds on S(T,x) for general cusp forms (on GL(d))
Recall that we expect S(T,x) =
e(nx) to be O
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
Main known result
S(T,x) = O
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
has infinitely many zeroes on its critical line
(we will see this again later).