Project III (MATH3382) 2020-21

Analysis meets Number Theory

P. Vishe

Description

The last few years have been very exciting 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. There has been something in common with all these advances: namely, that they use techniques from real and complex analysis to answer these questions. This field of study is usually coined as Analytic Number Theory. This project aims to study various analytic techniques that are used in number theory and their applications.

More explicitly, we aim to learn and to apply the following techniques:

  • Weyl's criterion for equidistribution;
  • Weyl's inequality and equidistribution of the sequence { n^2 y}, for an irrational y;
  • Exponential integrals and the saddle point method;
  • Euler Mclaurin 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.

    Prerequisite

     Elementary Number Theory II and Algebra II are essential.

    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.

    Walkden C. Lecture notes on Ergodic theory, Lecture VI, Manchester University Lecture notes

    email: Pankaj Vishe

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