  Building on work of Hardy and Ramanujan, Rademacher proved a well-known
formula for the values of the ordinary partition function $p(n)$. More
recently, Bruinier and Ono obtained an algebraic formula for these values. Here
we study the smallest parts function introduced by Andrews;
$\operatorname{spt}(n)$ counts the number of smallest parts in the partitions
of $n$. The generating function for $\operatorname{spt}(n)$ forms a component
of a natural mock modular form of weight $3/2$ whose shadow is the Dedekind eta
function. Using automorphic methods (in particular the theta lift of Bruinier
and Funke), we obtain an exact formula and an algebraic formula for its values.
In contrast with the case of $p(n)$, the convergence of our expression is
non-trivial, and requires power savings estimates for weighted sums of
Kloosterman sums for a multiplier in weight $1/2$. These are proved with
spectral methods (following an argument of Goldfeld and Sarnak).
