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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 power‐saving error term in the Titchmarsh divisor problem of estimating∑   p  ⩽  x    τ   (  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  ⩽  xτ  k    (  n  )   τ   (  n    +    1  )    , reproving a result announced by Bykovskiĭ and Vinogradov by a different method. The gain in the exponent is shown to be independent of   k   if a generalized Lindelöf hypothesis is assumed.

Sums of Kloosterman sums in arithmetic progressions, and the error term in the dispersion method