Abstract
for any e > 0. This estimate has had many applications, for example to sharpenings of the Brun-Titchmarsh theorem on primes in arithmetic progressions. Burgess' bound (2) is weaker than the trivial bound (1) for q < T 12. However there is an alternative method which improves upon (1) for sufficiently large one treats the q-dependence trivially and applies van der Corput's method to sums of the type
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