This part II introductory 10 credit year-three module uses elementary and combinatorial techniques to study how properties of divisibility can be used to characterise the integers. At the heart of the syllabus content is the topic of multiplicative arithmetic functions.

In number theory, an arithmetic, arithmetical, or number-theoretic function is a real or complex valued function f(n) defined on the set of natural numbers. We say that an arithmetic function which maps the natural numbers onto the complex plane, is multiplicative if f(mn) = f(m)f(n), for all natural numbers m and n with (m, n) = 1. If f(mn) = f(m)f(n) for all integers m and n, then we say that the function f is totally multiplicative. Dirichlet characters are totally multiplicative.

In addition to multiplicative functions, this module contains sections on division in congruences, solving equations modulo a prime power, Gaussian integers, representations of an integer as a sum or difference of two squares, roots of unity, Dirichlet Series, Dirichlet Characters and Mobius inversion.

Pre-requisite Modules: MA2011 Introduction to Number Theory I.

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• ·         Perform division in congruences
• ·         Solve congruences modulo a prime power
• ·         Solve power congruences
• ·         Factorise Gaussian Integers
• ·         Calculate the number of ways of expressing an integer n as a sum of two squares.
• ·         Calculate Dirichlet characters from the roots of unity
• ·         Define and manipulate the basic arithmetic functions
• ·         Apply Dirichlet’s convolution to Dirchlet series
• ·         Perform Mobius inversion

Modules will be delivered through blended learning. You will be guided through learning activities appropriate to your module, which may include:

• Weekly face to face classes (e.g. labs, lectures, exercise classes)
• Electronic resources that you work through at your own pace (e.g. videos, exercise sheets, lecture notes, e-books, quizzes)

Students are also expected to undertake self-guided study throughout the duration of the module.

Skills:
Working with congruences, division in congruences, congruences modulo a prime power, and Euler’s totient function. Understanding Gaussian factorisation and the numbers of ways of writing an integer as a sum of two squares. Familiarity with the basic arithmetic functions and ability to manipulate and evaluate them. Undertake Mobius inversion for Dirichlet Series and construct Dirichlet characters for a given modulus m.

Transferable Skills:
Recognising patterns. Reasoning logically. Solving equations modulo a prime power. Working with complex calculations and structures.

• ·          Congruences.
• ·          Division in congruences.
• ·          Congruences modulo primes.
• ·          Congruences modulo prime powers.
• ·          Algebraic Integers.
• ·          Gaussian factorisation
• ·          Dirichlet characters.
• ·          Sums of two squares.
• ·          Multiplicative functions
• ·          Dirichlet series and Dirichlet’s convolution.
• ·          Mobius's inversion.