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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Abstract.We determine a class of real valued, integrable functions fix) and corresponding functions MA[x) such that fix) < MA[x) for all x, the Fourier transform MA[t) is zero when |/| > 1, and the value of MA[0) is minimized. Several applications of these functions to number theory and analysis are given.

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Abstract. Let p n denote the n th prime. Goldston, Pintz, and Yıldırım recently proved that lim infWe give an alternative proof of this result. We also prove some corresponding results for numbers with two prime factors. Let q n denote the n th number that is a product of exactly two distinct primes. We prove that lim infIf an appropriate generalization of the Elliott-Halberstam Conjecture is true, then the above bound can be improved to 6.

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We show that using character sum estimates due to H. Iwaniec leads to an improvement of recent results about the distribution and finding RSA moduli M = pl, where p and l are primes, with prescribed bit patterns. We are now able to specify about n bits instead of about n/2 bits as in the previous work. We also show that the same result of H. Iwaniec can be used to obtain an unconditional version of a combinatorial result of W. de Launey and D. Gordon that was originally derived under the Extended Riemann Hypothesis.