MATH421 Analytic Number Theory

Overview

Number Theory is concerned with properties of integers, especially prime numbers. This paper gives a basic introduction to analytic number theory, which studies these objects using methods from analysis.

This course will provide you with the analytical tools and profound insights required to address advanced problems in number theory, establishing a robust foundation for further studies in this fascinating field of mathematics.

The paper starts with an exploration of the concepts of divisibility, factors, and prime numbers, leading to a comprehensive understanding of the fundamental theorem of arithmetic. Then we investigate multiple proofs of Euclid’s theorem about the infinitude of prime numbers, which serves as a cornerstone of number theory.

Our second topic will be the important class of arithmetic functions. In particular, we study Dirichlet multiplication, the Möbius inversion formula, generalised convolutions, and Legendre’s identity, a crucial tool in the study of arithmetic functions.

The third topic are asymptotic formulae. Gain proficiency in using the big O notation and understanding asymptotic equality. Learn Euler’s summation formula and Abel’s identity, valuable techniques for estimating infinite series. Then we explore Shapiro’s Tauberian theorem, a powerful tool for connecting asymptotic results to the properties of functions.

These preparations lay the foundation for investigating important properties of prime numbers. In particular, we prove the prime number theorem about the asymptotic distribution of prime numbers. We also study techniques for estimating the prime-counting function and the nth prime number. Of central importance will be Riemann's zeta function and an investigation of its zeros. For that purpose, we will employ methods from complex analysis.

About this paper

Restriction

MATH 401-412

Limited to

BA(Hons), BSc(Hons), PGDipArts, PGDipSci, MA (Thesis), MSc, MAppSc, PGDipAppSc, PGCertAppSc

Contact

Mathematics 400-level programme coordinator: Dr Fabien Montiel fabien.montiel@otago.ac.nz

Teaching staff

Dr Joerg Hennig

Paper Structure

Main topics (18 lectures, 50 min each):

Fundamentals:

• Divisibility, factors, prime numbers.
• Fundamental theorem of arithmetic.
• Various proofs of Euclid’s theorem.

Arithmetic functions:

• Dirichlet multiplication.
• The Möbius inversion formula.
• Generalised convolutions.
• Legendre’s identity.

Asymptotic formulae:

• Big O notation and asymptotic equality.
• Euler’s summation formula and Abel’s identity.
• Elementary asymptotic formulae.
• Shapiro’s Tauberian theorem.

The distribution of prime numbers:

• The prime number theorem.
• Equivalent formulations.
• Bounds for the prime-counting function and the nth prime.

Riemann’s zeta function:

• Analytic continuation of the zeta function.
• Functional equation.
• Trivial and nontrivial zeros.

Teaching Arrangements

18 lectures (50 minutes each).

Tutorials: 3 tutorials (50 min) to provide help with each of the 3 assignments.

Assessment: 3 written Assignments (40%), final exam (60%)

Graduate Attributes Emphasised

Critical Thinking, Interdisciplinary Perspective, Communication, Information Literacy, Lifelong Learning.
View more information about Otago's graduate attributes.

Learning Outcomes

On completion of the study of this paper, students are expected to:

• Understand fundamental approaches in analytic number theory.
• know important properties of prime numbers and their distribution.
• Be able to derive asymptotic formulae.
• Understand how methods from real and complex analysis can be used to rigorously prove theorems in number theory.
• Know basic properties of Riemann’s zeta function, get an impression of the large number of difficult open problems in analytic number theory.

Timetable