Summation
Formulas

•Recall the Voronoi summation formula for GL(2): if

–f(x) is a Schwartz function which vanishes to infinite
order at the origin

–an are the coefficients of a modular or Maass
form for SL(2,Z)

–a, c relatively prime integers,

then

where

•This formula has many analytic uses for dualizing sums
of coefficients (e.g.
subconvexity, together with trace formulas).

•It can be derived from the standard L-function (if a=0),
and from its twists
(general a,c). The usual proofs involve
special functions, but the
final answer does not. Is that
avoidable?