Analytic Number Theory

Lecturer:

Dr Michael Coons, The University of Newcastle

Synopsis

Questions in number theory have driven the main stream of mathematical research for millennia, producing avenues that leading to significant advancement in several other areas of mathematics. Stemming from the interest in the arithmetic of the integers one encounters the basic operations of ‘plus’ and ‘times’. While ‘plus’ is easy enough, the intricacies of ‘times’ continue to be an object of rigorous mathematical study. The first important result in this vein, is the proof of the infinitude of the primes—the multiplicative building blocks of the integers—recorded by Euclid in his Elements.

In this course, we will study the distribution of primes in it’s classical development before moving onto other important functions in the context of the multiplicative properties of integers.

Course Overview

  • Infinitude of primes
  • Chebyshev’s functions
  • Bertrand’s theorem
  • Partial summation
  • Riemann’s zeta function
  • Prime number theorem
  • Newman’s theorem
  • Dirichlet L-functions
  • Primes in arithmetic progressions

Given time we may cover:

  • Primitive roots and quadratic reciprocity
  • Sums of multiplicative functions and density
  • Roth’s theorem and the circle method
  • Vinogradov’s three prime theorem

Prerequisites:

A course on complex analysis is a must as well as competence with proofs and the basic concepts of real analysis. A course in elementary number theory would be helpful, but is certainly not required for the motivated student.

Assessment

  • Assignments: 40%
  • Final Examination: 60%

Resources

No book will be required. PDF file(s) of course notes will be made available to the student.

Analytic Number Theory

Biography

Dr Michael Coons, University of Newcastle

Michael is Senior Lecturer at the University of Newcastle. Previously, he worked at the Fields Institute and the University of Waterloo, Canada. Since coming to Australia in 2012, he has been an Australian Research Council DECRA Fellow, a Visiting Researcher at the Alfréd Rényi Institute of the Hungarian Academy of Sciences in Budapest, Hungary, and a Visiting Professor at the Centre for Mathematical Modelling of the University of Bielefeld, Germany. Michael’s research interests lie in number theory as broadly defined, with connections to several other areas of pure mathematics including algebra, analysis, combinatorics, dynamical systems and theoretical computer science.

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