University of Illinois at Urbana-Champaign
The lectures will focus on multiplicative number theory. Topics to be selected from following list, depending on time and audience preparation and interest.
1. Elementary theory of multiplicative functions. Convolutions
3. Analytic theory. Dirichlet series, Euler products, applications
5. Mean values. Elementary theory, Halasz theorem
6. Numbers having only small prime factors
A prose description of the topics:
An arithmetic function f 0 is called multiplicative if it satisfies the relation f(mn) = f(m) f(n) for all relatively prime positive integers m,n. This modest requirement imposes significant structure on an arithmetic function, and many interesting functions are either multiplicative or are `nearly so.' We are going to examine several aspects of multiplicative functions, including the following.
Some famous number theoretic questions such as the prime number theorem, the Dirichlet divisor problem, and the distribution of square-free numbers will be considered in terms of multiplicative functions.
Multiplicative functions will be characterized in terms of convolutions and exponentials of arithmetic functions. In particular, these functions will be shown to form a group under convolution.
Associated with each arithmetic function is a Dirichlet series, which can provide useful analytic information about the function. For multiplicative functions the Dirichlet series admits a representation in factored form, the so-called Euler product. Several examples of this relation will be studied, including the world's most famous Dirichlet series, the Riemann zeta function.
The summatory function associated with a multiplicative function is often of greater interest than the function itself. Some elementary and analytic techniques will be presented for estimating such functions. In particular, we shall ask Which multiplicative functions have a mean value? Theorems of Delange and Halasz will be presented which give conditions for a mean value.
shall consider the counting function of integers having no
small prime factors. The Dickman function will be introduced
and its properties examined.