Explicit Zero Density for the Riemann zeta function

The Riemann zeta function is a fundamental function in number theory. The study of zeros of the zeta function has important applications in studying the distribution of the prime numbers. Riemann hypothesis conjectures that all non-trivial zeros lie on the critical line, while the trivial zeros occur at negative even integers. A less ambitious goal than proving there are no zeros is to determine an upper bound for the number of non-trivial zeros, denoted as $N(\sigma,T)$, within a specific rectangular region defined by $\sigma < \Re{s} < 1$ and $0< \Im{s} < T$. Previous works by various authors like Ingham and Ramare have provided bounds for $N(\sigma,T)$. In 2018, Habiba Kadiri, Allysa Lumley, and Nathan Ng presented a result that provides a better estimate for $N(\sigma,T)$. In this talk, I will give an overview of the method they provide to deduce an upper bound for $N(\sigma,T)$. My thesis will improve their upper bound and also update the result to use better bounds on $\zeta$ on the half line among other improvements.