This book is a self-contained account of the one- and two-dimensional van der Corput method and its use in estimating exponential sums. These arise in many problems in analytic number theory. It is the first cohesive account of much of this material and will be welcomed by graduates and professionals in analytic number theory. The authors show how the method can be applied to problems such as upper bounds for the Riemann-Zeta function. the Dirichlet divisor problem, the distribution of square free numbers, and the Piatetski-Shapiro prime number theorem.

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Let H be a Hardy field (a field consisting of germs of real-valued functions at infinity that is closed under differentiation) and let f ∈ H be a subpolynomial function. Let P = {2, 3, 5, 7, . . . } be the (naturally ordered) set of primes. We show that (f (n)) n∈N is uniformly distributed mod 1 if and only if (f (p)) p∈P is uniformly distributed mod 1. This result is then utilized to derive various ergodic and combinatorial statements which significantly generalize the results obtained in [BKMST].

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