We prove a bound for quintilinear sums of Kloosterman sums, with congruence
conditions on the "smooth" summation variables. This generalizes classical work
of Deshouillers and Iwaniec, and is key to obtaining power-saving error terms
in applications, notably the dispersion method. As a consequence, assuming the
Riemann hypothesis for Dirichlet L-functions, we prove a power-saving error
term in the Titchmarsh divisor problem of estimating ∑p≤xτ(p−1).
Unconditionally, we isolate the possible contribution of Siegel zeroes, showing
it is always negative. Extending work of Fouvry and Tenenbaum, we obtain
power-saving in the asymptotic formula for ∑n≤xτk(n)τ(n+1),
reproving a result announced by Bykovski\u{i} and Vinogradov by a different
method. The gain in the exponent is shown to be independent of k if a
generalized Lindel\"of hypothesis is assumed