MATH 788T Elementary Methods in Analytic Number Theory Course Description

Most of the central results in Analytic Number Theory like the Prime Number Theorem were first proved via complex variables. However, an important part of Analytic Number Theory is finding elementary proofs of classical results. For instance, the discovery of an elementary proof of the Prime Number Theorem, done independently by Selberg and Erdős is one of the greatest achievments in Number Theory. In the last few years a number of elementary methods were developed that lead to results that are currently out of reach of any other methods. In this course we discuss some of the new elementary methods and their applications. The central theme will be techniques for estimating the number of integer and rational points close to a smooth curve with applications to the distribution of squarefree and squarefull numbers in short intervals. Also, we will discuss elementary aspects of the Dirichlet Divisor problem and the Gauss Circle Problem.

Some of the results we will obtain are:

1) There exists a constant c > 0 such that the interval (x, x+cx^{\frac{1}{5}} log x] contains a squarefree number for every x ≥ 1.

2) Asymptotic formula for the number of squarefull numbers in a short interval.

3) A number of estimates for the number of integer points close to a smooth curve.

4) Estimates of the least prime divisor of a binomial coefficient.

5) Results about the relation between the area of a two-dimensional domain and the number of lattice points it contains.

The discussion of the above results will lead us in a natural way to a number of other topics, including properties of Farrey fractions, summation formulas, divided differences, polynomial identities, and lattices in vector spaces.