Van der Corput’s method for exponential sums. (English) Zbl 0713.11001

Graham, S. W.; Kolesnik, G.

London Mathematical Society Lecture Note Series, 126. Cambridge etc.: Cambridge University Press. 120 p. £11.95; $ 22.95 (1991).

In this monograph simple and double exponential sums of types \[ \sum_{a<n\leq b}e^{2\pi i f(n)},\quad \sum_{(m,n)\in D}e^{2\pi i f(m,n)} \] are considered. Such exponential sums play an important role in the theory of Riemann zeta function, in problems of asymptotic results for mean values of arithmetic functions and in counting the number of lattice points in large regions. Van der Corput’s method of estimating simple exponential sums and recent results on double exponential sums are developed.
The text book consists of seven chapters. In the first introductory chapter a historical overview is given. In Chapter 2 the simplest van der Corput estimate is proved. In Chapter 3 the method of exponent pairs is described. In Chapter 4 some applications to important classical problems are considered. The Graham algorithm of computing optimal exponent pairs is developed in Chapter 5. In Chapter 6 there are some results on double exponential sums and Kolesnik’s AB-Theorem. Chapter 7 is devoted to Bombieri’s and Iwaniec’s new method of estimating simple exponential sums and the extension by Huxley and Watt.
This is a short but highly instructive introduction into one of the most important methods of modern analytic number theory.

Reviewer: Ekkehard Krätzel (Wien)

MSC:

11-02	Research exposition (monographs, survey articles) pertaining to number theory
11L07	Estimates on exponential sums
11M26	Nonreal zeros of \(\zeta (s)\) and \(L(s, \chi)\); Riemann and other hypotheses
11N37	Asymptotic results on arithmetic functions
11P21	Lattice points in specified regions

Keywords:

Riemann zeta function; asymptotic results for mean values of arithmetic functions; number of lattice points; Van der Corput’s method; Graham algorithm; exponent pairs; double exponential sums; Kolesnik’s AB-Theorem