The Dirichlet Divisor Problem, named after the German mathematician Peter Gustav Lejeune Dirichlet, is a classical problem in number theory. It concerns the distribution of the number of divisors of positive integers, which plays a crucial role in various areas of mathematics, including analytic number theory and algebraic geometry.

The problem can be succinctly stated as follows: given a positive integer n, let the divisor function d(n) be the number of positive integer divisors of n (including 1 and n itself). From now on, we only deal with positive integers unless otherwise specified. If we let D(x) sum the numbers of divisors of integers less than or equal to x. Thus, the average number of divisors for any integer smaller or equal to x is D(x)/x. In 1849, Dirichlet proposed the question of the size of D(x) and developed a method, now known as the hyperbola method, to prove that 𝐷(𝑥)=∑𝑛≤𝑥𝑑(𝑛)=𝑥log𝑥+𝑥(2𝛾−1)+𝑂(ξ𝑥), where𝛾=.577215… is Euler’s constant. The problem of finding the best error term in the expression for D(x) is now known as the Dirichlet divisor problem. Despite its seemingly elementary nature, the Dirichlet divisor problem remains one of the most challenging unsolved problems in number theory. Progress has been made in certain special cases and under various assumptions, but a complete understanding of its behavior remains elusive.

It turns out the error term is closely connected to the estimated of exponential sums which can be studies using Van der Corput’s method.

In this paper, we will introduce Van der Corput’s method of exponential sums, which provides a systematic way to bound exponential sums by exploiting the oscillatory behavior of the complex exponential function. We will describe Van der Corput’s method as well as its application to the Dirichlet divisor problem.