UNIVERSITY OF WAIKATO
Department of Mathematics
MATH515-12B Ė Analytic Number Theory - 2012
web site: http://http://www.math.waikato.ac.nz/~kab/math515-12B.html
For some time now there has been developing within and outside of mathematics a renewed energy and interest in matters relating to number theory. This has come from not only the recent solution to some unsolved long outstanding problems using modern methods, like Wilesí proof of Fermatís Last Theorem, but also because of the pressing need for effective encryption in commercial, strategic and personal computer based communications. In addition, the use of the computer has made it possible to explore a much wider domain of number based phenomena than before, leading to new ideas. However, quite a few classical problems relating to the fundamental structure of natural numbers, are still unsolved. Since these number underpin all of mathematics and its applications, there is a lot of challenge and life left in this subject.
The following is a list of the type of topics which might be included, but it is not exhaustive and all topics listed would not necessarily be covered:
- Summation flormulae and inequalities
- Prime numbers and their properties
- The Riemann zeta function
- The proof of the prime number theorem
- The Euler phi function
- The ABC conjecture
- Prime numbers in arithmetic progression
- Characters and Dirichlet's theorem
- The Hardy-Ramanujan and Landau theorems
- The global and local behaviour of arithmetic functions
By Friday 5pm each week you should hand in (through the labelled slot outside the mathematics office) the solution to one problem which I will set from the text or in lectures. Tutorial participation will amount to 20% of the assessment. There will also be a final 2 hour test, mostly bookwork (the content will be prescribed) during the examination period. A full pass in the paper requires a D or better in the final test. For assessment purposes the 10 best assignments, out of 11, will be included, and the weighting assignments plus tutorials to test will be 1:1.
Kevin Broughan, 18th July 2012.