  The original formulae of Kuznetsov for $SL(2,\mathbb{Z})$ allowed one to
study either a spectral average via Kloosterman sums or to study an average of
Kloosterman sums via a spectral interpretation. In previous papers, we have
developed the spectral Kuznetsov formulae at the minimal weights for
$SL(3,\mathbb{Z}))$, and in these formulae, the big-cell Kloosterman sums occur
with weight functions attached to four different integral kernels, according to
the choice of signs of the indices. These correspond to the $J$- and $K$-Bessel
functions in the case of $GL(2)$. In this paper, we demonstrate a linear
combination of the spherical and weight one $SL(3,\mathbb{Z})$ Kuznetsov
formulae that isolates one particular integral kernel, which is the spherical
$GL(3)$ Whittaker function. Using the known inversion formula of Wallach, we
give the first arithmetic Kuznetsov formula for $SL(3,\mathbb{Z})$ and use it
to study smooth averages and the Kloosterman zeta function attached to this
particular choice of signs.

Number Theory