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Science Mathematics and Statistics

MATH5515 Special Topics in Pure Mathematics A

Overview

MATH5515 is a Honours and Postgraduate coursework Mathematics course. The Topic title is聽Analytic Number Theory.聽 聽Complex analysis will be a pre-requisite for this course and all other tools will be developed as needed.

Units of credit:听6

Prerequisites:聽MATH2521 or MATH2621 Complex Analysis.

Cycle of offering:聽Term 1聽

Graduate attributes:听The course will enhance your research, inquiry and analytical thinking abilities.聽

More information:听 聽The Course outline will be made available closer to the start of term - please visit this website:听www.unsw.edu.au/course-outlines聽 聽 聽

Analytic Number Theory topics will include:听

路聽聽聽聽聽聽 The Riemann zeta-function, arithmetic functions and Dirichlet Series.

路聽聽聽聽聽聽 Analytic Properties of the zeta-function and Functional Equation, Poisson Summation Formula, Properties of the Gamma function.

路聽聽聽聽聽聽 Integral Functions of Order 1, the Hadamard Product.

路聽聽聽聽聽聽 The Gamma function.

路聽聽聽聽聽聽 Zero-free Regions of the zeta-function.

路聽聽聽聽聽聽 Distribution of complex zeros of the zeta-function.

路聽聽聽聽聽聽 The Explicit Formula.

路聽聽聽聽聽聽 The Prime Number Theorem.

路聽聽聽聽聽聽 Hardy's Theorem.

路聽聽聽聽聽聽 Primes in Arithmetic Progressions

Important additional information as of 2023

UNSW Plagiarism Policy

The University requires all students to be aware of its聽.

For courses convened by the聽School of Mathematics and Statistics no assistance using generative AI software is allowed unless specifically referred to in the individual assessment tasks.

If its use is detected in the no assistance case, it will be regarded as serious academic misconduct and subject to the standard penalties, which may include 00FL, suspension and exclusion.

The entry聽contains聽information about the course. (The timetable is only up-to-date if the course is being offered this year.)

If you are currently enrolled in MATH5515, you can log into聽聽for this course.

Course overview

This is an introductory course on the Riemann zeta-function, in which the methods and results of complex analysis will be used extensively. We will build our knowledge of this function and arrive at the prime number theorem. Time permitting, a number of other important results will be introduced as well. The following is a list of topics that will be covered:
  • The Riemann zeta-function
  • Entire Functions of Finite Order
  • The Gamma Function
  • Distribution of the Complex Zeros
  • The Explicit Formula
  • The Zero-Free Region
  • The Prime Number Theorem
  • Hardy's Theorem
  • Primes in Arithmetic Progressions