•Let an be the
coefficients of a cusp form L-function on GL(d):

S(T,x) = ån6T an e(n x) , e(t) := e 2 p i t

• Since the an have unit size on
average, we have the following two trivial bounds:

– S(T,x) = O(T)

– sR/Z |S(T,x)|2 dx = ån6T |an|2 ~ cT

•__Folklore Cancellation Conjecture__: S(T,x) = Oe(T1/2+e), where the implied
constant depends e but is uniform in x and T.

•In light of the L2-norm
statement, this is the best possible exponent.

Cancellation
in sums with additive twists