Project IV (MATH4072) 2016-17

Exponential sums, Exponential integrals and Applications

P. Vishe

Description

The last few years have been quite exciting for analytic number theorists with giant leaps made in various open problems in the field. For example, the results like the proof of ternary Goldbach conjecture, Zhang, Maynard, Tao's work on small gaps between primes, results on existence of rational points on varieities have taken a center stage in number theory. Theory of exponential sums and integrals has played a key role in all these works. The key aim of this project is to learn various techniques in bounding exponential sums and exponential integrals, and applications to problems in number theory. One such application is to the equidistribution problems in number theory, for example, Weyl's criterion for equidistribution of an arbitrary sequence of real numbers { x_n modulo 1}.

During the project we aim to learn and to apply the following techniques:

  • Weyl's criterion for equidistribution of a sequence modulo 1;
  • Weyl's inequality and equidistribution of the sequence { n^2 y}, for an irrational y;
  • Exponential integrals and the saddle point method;
  • Euler-Maclaurin summation formula;
  • Some other possible applications e.g. to the bounds for zeta function, Waring's problem etc.

    These techniques are quite standard and yet very crucial in analytic number theory. Some of them involve good understanding of tools from real and complex analysis and elementary number theory. We will review some of the pertinent material in the first term.

    Prerequisites

    2H core modules. Complex Analysis II, Elementary Number Theory II and Algebra II are essential. The Number Theory IV module will help.

    Resources

    Davenport H. Analytic Methods for Diophantine Equations and Diophantine Inequalities, (Chapter 3), Cambridge university press, ISBN: 9780521605830.

    Editors: Granville A., Rudnick Z. Equidistribution in Number Theory, An Introduction, (Pages 1-11), Springer, Nato science series, volume 237.

    Ivić A. The Riemann Zeta-Function, (chapter 2), John Wiley & Sons, New York, 1985.

    Tenenbaum G. Introduction to Analytic and Probabilistic Number Theory, (Chapter I.0), Cambridge Studies in Advanced Mathematics, vol. 46, Cambridge University Press, Cambridge, 1995.

    • email: Pankaj Vishe

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