Zero-Density Estimates for the Riemann Zeta Function
The Riemann zeta function is one of the most extensively studied functions in pure mathematics. This is for good reason: the theory surrounding this function has many profound implications across analytic number theory and beyond. In particular, it is deeply connected with the prime numbers. While initially introduced by Euler, it was Riemann who extended this function to a complex variable and thereby discovered the relation between its zeros and the distribution of primes in his revolutionary paper from 1859. At the same time, he formulated his most famous conjecture, which later became known as the Riemann hypothesis. Namely, he suggested that all non-trivial zeros of the zeta function lie on the so-called critical line, i.e. the line in the complex plane with real part 1/2. This problem has remained unsolved for over 150 years, with few major breakthroughs, and mathematicians still have no clear idea of where to look for a solution. Fortunately, over the past century, mathematicians have gained the insight that the full generality of the Riemann hypothesis is not always required in order to deduce good arithmetic consequences. Notably, in the search for zeros of the zeta function, special attention has been given to zero-density estimates. Rather than improving known zero-free regions, this subject aims to bound the number of zeros in certain rectangles inside the critical strip, the region of the complex plane consisting of all elements with real part between 0 and 1. This has become a central topic in analytic number theory, with several applications to prime number theory. In this thesis, our aim is to provide a thorough account of the techniques used to prove zero-density estimates. This begins with the description of the standard zero-detection method, and proceeds with two examples, one of which is the current record in a certain interval. We then exploit recent advances in the field to prove a new zero-density estimate that improves the current best bounds in a small interval, effectively breaking a world record. An essential tool in proving zero-density estimates is the study of moment estimates for the Riemann zeta function on the critical line, as they arise naturally in the zero-detection method. This subject has plenty of unsolved problems itself, and we will prove some of the current best results. The thesis concludes with an application of zero-density estimates to the theory of primes in short intervals.
Meer lezen
