Rationals vs.
Irrationals

•Fix x 2 **Q****.** S(T,x) can be smaller = Ox(T1/2-d)
(Landau).

–For example, the sum S(T,0) = ån6T an is typically
quite small, because for
example:

•L(s) = ån>1 an n-s is entire

•Smoothed sums behave even better:

decays rapidly in T (faster than any polynomial), for y say a Schwartz function on (0,1).

[shift
contour s to -1]

–Similar behavior at other rationals (related to
L-functions twisted by Dirichlet
characters).

•However, *uniform* bounds
over rationals x are still not easy.