web site: http://http://www.math.waikato.ac.nz/~kab/math515-12B.html

Rationale:

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 Wiles’ proof of Fermat’s 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.

Details of the paper content:

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:

1. Summation flormulae and inequalities
2. Prime numbers and their properties
3. The Riemann zeta function
4. The proof of the prime number theorem
5. The Euler phi function
6. The ABC conjecture
7. Prime numbers in arithmetic progression
8. Characters and Dirichlet's theorem
9. The Hardy-Ramanujan and Landau theorems
10. The global and local behaviour of arithmetic functions

Bibliography:

1. An introduction to the Theory of Numbers G H Hardy and E M Wright (Oxford 1938-1999)
2. Introduction to Analytic Number Theory, T M Apostol (Springer Verlag 1976)
3. Not always burried deep, P Pollack (AMS)
4. Elementary methods in number theory, M Nathanson (Springer)

Web Sources: