Dr. Stephen Lucas, 18 September, 2006
 A Direct Proof of the Prime Number Theorem
Abstract:  A well known theorem of number theory is the prime number theorem, which states that the number of prime numbers less than or equal to some integer x is asymptotically equal to x/log(x). This talk will begin by summarizing the history of the prime number theorem. Then, we describe a transform which is up to a change of variable a Laplace transform, and use it to find a relationship between the average order of an arithmetic function and its Dirichlet series. The singularities of a function of the Dirichlet series can be used to find the asymptotic form of the average order, a generalization of Ikehara's theorem. We use this, along with the properties of a function related to the Riemann zeta- function, to form a new proof of the prime number theorem. The same technique is also used to find both Riemann's approximation to the number of primes and an exact summation. This talk assumes no previous knowledge of number theory, and only an undergraduate knowledge of transform theory. We conclude with an intriguing possibility for extending this approach to other important problems.

Biographical Sketch:  Steve Lucas received a 1st Class honours degree with University Medal in Mathematics from Wollongong University in 1989, and a PhD in Mechanical Engineering from Sydney University in 1994. He did a Postdoctoral Research Fellowship in the Division of Applied Sciences, Harvard University during the period 1993-1995. From 1996-2006 he was employed by the School of Mathematics in the University of South Australia, including a visiting position 2005-2006 at the Department of Mathematics, University of Tennessee. He is currently a visiting associate professor within the Department of Mathematics and Statistics, James Madison University. Apart from number theory, Steve's research interests include analysis, fluid and solid mechanics, signal processing, control theory, graph theory and industrial mathematics, but the majority of his research has a numerical flavor. In February 2002, Steve received the Michell Medal for the outstanding new researcher in Applied Mathematics in Australia and New Zealand, awarded by the Australia New Zealand Industrial and Applied Mathematics division of the Australian Mathematical Society.