Arthur's trace formula and residues of Dedekind zeta functions

Arthur's trace formula for a reductive group G is an important tool in
number theory. We shall introduce a large natural class of non-compactly
supported test functions for which there is an absolutely convergent
expansion of the spectral side of the trace formula by a recent result of
Finis-Lapid-Müller, and for the geometric side for GL(2) by work of
Finis-Lapid. This allows us to use particular test functions for GL(2) to
recover classical results on the asymptotic behaviour of mean values of
residues of Dedekind zeta functions of quadratic fields. We can find an
absolutely convergent expansion for the geometric side for GL(3) as well,
and hence shall see what asymptotics we have and should eventually get in
this case.