Faculty of Science School of Mathematics and Statistics MATH5645 NUMBER THEORY Semester 1, 2014 CRICOS Provider No: 00098G MATH5645 – Course Outline Information about the course Course Authority: P.G. Brown Lecturer: P.G. Brown RC-3055, email [email protected]. Consultation: Please use email to arrange an appointment. Credit, Prerequisites, Exclusions: This course counts for 6 Units of Credit (6UOC). MATH2620/MATH3611 is assumed knowledge for this course. Lectures: There will be three lectures per week at times to be decided. Moodle: Further information, skeleton lecture notes, and other material will be provided via Moodle. Course aims This course aims to examine some of the key ideas and techniques of Analytic Number Theory, and to work through in some detail the proofs of the famous Prime Number Theorem and Dirichlet’s Theorem. Relation to other mathematics courses Mathematics may be divided into the broad categories of analysis (calculus), algebra, geometry and logic. This subject uses ideas from analysis and algebra to tackle problems in Number Theory. Student Learning Outcomes Students taking this course will gain an understanding of the basics of Analytic Number Theorey. They will: • understand the key problems relating to the distribution of prime numbers; • appreciate how analytic tools can be introduced to solve such problems; • examine the basic properties of the arithmetic functions and their asymptotic behaviour; 2 Through regularly attending lectures and applying themselves in tutorial exercises, students will reach the above outcomes. Relation to graduate attributes The above outcomes are related to the development of the Science Faculty Graduate Attributes, in particular: 1.Research, inquiry and analytical thinking abilities, 4. Communication, 6. Information literacy Teaching strategies underpinning the course New ideas and skills are introduced and demonstrated in lectures, then students develop these skills by applying them to specific tutorial and assessment questions. Rationale for learning and teaching strategies We believe that effective learning is best supported by a climate of enquiry, in which students are actively engaged in the learning process. To ensure effective learning, students should participate in class as outlined below. We believe that effective learning is achieved when students attend all classes, have prepared effectively for classes by reading through previous lecture notes, in the case of lectures, and, in the case of tutorials, by having made a serious attempt at doing for themselves the tutorial problems prior to the tutorials. Furthermore, lectures should be viewed by the student as an opportunity to learn, rather than just copy down lecture notes. Effective learning is achieved when students have a genuine interest in the subject and make a serious effort to master the basic material. Assessment Assessment in this course will consist of six short assignments counting a total of 30%, a tutorial mark worth 4% and a final examination (66%) covering all of the course. Assessment criteria: The main criteria for marking all assessment tasks will be clear and logical presentation of correct solutions. Unnecessarily complicated and obstruse solutions will be marked down. 3 Assignments Rationale: Assignments will give an opportunity for students to try their hand at problems requiring more than one line of argument and also introduce them to aspects of the subject which are not explicitly covered in lectures. Each assignment is due at the last lecture for the indicated week. MATH5645 Assignment Schedule. Assignments will be due in Weeks 3,5,7,9,11 and 13. Each will consist of only 3 questions. Assignments must be YOUR OWN WORK, or severe penalties will be incurred. I am happy to give hints if you are unsure how to begin a problem. You should consult the University web page on plagiarism www.lc.unsw.edu.au/plagiarism Late assignments will not be accepted. Tutorial Presentation In the last lecture time for each topic we will have a tutorial. Students will be divided into groups and each group will be responsible for presenting some of the solutions to the tutorial problems for that section. Further details and times for these will be given in lectures. Examination Duration: Three hours. Rationale: The final examination will assess student mastery of the material covered in the lectures. Weighting: The final examination will count for 66% of your final mark. Further details about the final examination will be available in class closer to the time. Additional resources and support Tutorial Exercises A set of tutorial exercises will be available through Moodle. These problems are for YOU to do to enhance mastery of the course. 4 Lecture notes A set of skeleton lecture notes for the course will appear on Moodle. Textbooks There is no set text for this course. The content of the course will be defined by the lectures. The following books (in no particular order) might be useful: Introduction to Analytic Number Theory, T.M. Apostol, 1980 edition Prime Numbers, W. and F. Ellison, 1985 Problems in Analytic Number Theory, M. Ram Murty, (GTM), 2001. A Course in Number Theory, H.E. Rose, 1988 The Riemann Zeta Function, A.A. Karatsuba and S.M. Voronin, 1992. An Introduction to the Theory of Numbers, G.H. Hardy and E.M. Wright, 1959 (Fourth edition). The Development of Prime Number Theory, W. Narkiewicz, 2000 An Introduction to Arithmetical Functions, P. J. McCarthy, 1986. The Prime Number Theorem, G.J.O. Jameson, 2003 A Primer of Analytic Number Theory, J.Stopple, 2003 An Invitation to Modern Number Theory, S.J. Miller and R. Takloo-Bighash, 2006. The Prime Number Theorem, G.J.O. Jameson, 2003. Course Evaluation and Development The School of Mathematics and Statistics evaluates each course each time it is run. We carefully consider the student responses and their implications for course development. It is common practice to discuss informally with students how the course and their mastery of it are progressing. 5 Administrative matters Additional Assessment See attached handout. School Rules and Regulations Fuller details of the general rules regarding attendence, release of marks, special consideration etc are available via the School of Mathematics and Statistics Web page at http://www.maths.unsw.edu.au/students/current/policies/studentpolicy.html. Plagiarism and academic honesty Plagiarism is the presentation of the thoughts or work of another as one’s own. Issues you must be aware of regarding plagiarism and the university’s policies on academic honesty and plagiarism can be found at http://www.lc.unsw.edu.au/plagiarism and http://www.lc.unsw.edu.au/plagiarism/plagiarism_STUDENTBOOK.pdf. Course Schedule: Here is a rough guide as to the schedule of material taught. It is planned to spend two weeks on each section, however, if we get behind, the last section may be deleted. 1. Some elementary number theory. This includes the theorems of Fermat, Euler, Wilson, quadratic reciprocity, sums of squares, Waring’s problem. 2. The Euler product, Bertand’s Postulate, primes in certain arithmetic progressions and Chebychev’s inequalities. 3. The Arithmetic Functions and their asymptotics, Dirichlet multiplication. 4. Abelian Characters, Dirichlet L-series and Dirichlet’s Theorem. 5. The Riemann Zeta Function. 6. The Prime Number Theorem. 6