  We consider a problem posed by Shparlinski, of giving nontrivial bounds for
rational exponential sums over the arithmetic function $\tau(n)$, counting the
number of divisors of $n$. This is done using some ideas of Sathe concerning
the distribution in residue classes of the function $\omega(n)$, counting the
number of prime factors of $n$, to bring the problem into a form where, for
general modulus, we may apply a bound of Bourgain concerning exponential sums
over subgroups of finite abelian groups and for prime modulus some results of
Korobov and Shkredov.
