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. It also touches on some special numbers, such as Bernoulli and Euler numbers, and their corresponding polynomials. 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.

In addition to multiplicative functions, this module contains sections on division in congruences, solving equations modulo a prime power, quadratic reciprocity, roots of unity, Dirichlet Series, Dirichlet Characters and Mobius inversion.

Pre-requisite Module: MA2011 Introduction to Number Theory I or MA0216 Elementary Number Theory II

• Perform division in congruences

• Solve congruences modulo a prime power

• Solve power congruences

• Calculate Dirichlet characters from the roots of unity

• Calculate the number of ways of expressing an integer n as a sum of two squares.

• Apply Jacobi’s algorithm

• Define and manipulate the basic arithmetic functions

• Apply Dirichlet’s convolution to Dirchlet series

• Perform Mobius inversion

• Derive Bernoulli and Euler polynomial identities

22 lectures (50 minutes each)

5 example classes (50 minutes each)

Some handouts and lecture notes will be provided in hard copy or via Learning Central, but students will be expected to take notes of lectures.

Students are also expected to undertake at least 50 hours private study including preparation of worked solutions to formative exercise sheets.

Skills:
Working with congruences, division in congruences, congruences modulo a prime power, and Euler’s totient function. Understanding quadratic reciprocity and applying Jacobi’s algorithm. 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.

Formative assessment is carried out in the examples classes.  Feedback to students on their solutions and their progress towards learning outcomes is provided during these classes.  

The summative assessment is the written examination at the end of the module.  This gives students the opportunity to demonstrate their overall achievement of learning outcomes.  It also allows them to give evidence of the higher levels of knowledge and understanding required for above average marks. 

The examination paper has a choice of three from four equally weighted questions.

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  Division in congruences.

• Congruences modulo primes.

• Congruences modulo prime powers.

• Algebraic Integers.

• Dirichlet characters.

• Sums of two squares.

• Quadratic reciprocity.

• Jacobi's algorithm.

• Multiplicative functions

• Dirichlet series and Dirichlet’s convolution.

• Mobius's inversion.

• Bernoulli and Euler numbers and polynomials.

Jones, G.A. and Jones, J.M. 1998. Elementary number theory. Springer.

Rosen, K. H. 2011. Elementary number theory and its applications. 6th ed. McGraw Hill 

Hardy, G.H. & Wright, E.M. 2008. Introduction to the theory of numbers. 6th ed. OUP – highly recommended.

Davenport, H. 2008. The higher arithmetic. 8th ed. Cambridge – elementary.