Integer points, exponential sums and the Riemann zeta function. (English) Zbl 1030.11053

Huxley, M. N.

Bennett, M. A. (ed.) et al., Number theory for the millennium II. Proceedings of the millennial conference on number theory, Urbana-Champaign, IL, USA, May 21-26, 2000. Natick, MA: A K Peters. 275-290 (2002).

This article gives a short but very nice survey of the development of estimating exponential sums and their applications to lattice point theory and the theory of the Riemann zeta-function. It begins with the handling of planar problems by Pfeiffer in the 19th century and ends with the discrete circle method by Bombieri and Iwaniec.
The brand-new estimates in the circle, divisor and Lindelöf problems are presented. Finally, a sketch of the proof of new results on the short interval mean square of exponential sums and the mean square of the Riemann zeta-function is given.
For the entire collection see [Zbl 1002.00006].

Reviewer: Ekkehard Krätzel (Wien)

MSC:

11P21	Lattice points in specified regions
11L07	Estimates on exponential sums
11-02	Research exposition (monographs, survey articles) pertaining to number theory
11M06	\(\zeta (s)\) and \(L(s, \chi)\)

Keywords:

lattice points; estimates on exponential sums; Riemann zeta-function