Lethbridge Number Theory and Combinatorics Seminar: Golnoush Farzanfard

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(σ,T), within a specific rectangular region defined by σ<Rs<1 and 0<Ims<T . Previous works by various authors like Ingham and Ramare have provided bounds for N(σ,T). In 2018, Habiba Kadiri, Allysa Lumley, and Nathan Ng presented a result that provides a better estimate for N(σ,T). In this talk, I will give an overview of the method they provide to deduce an upper bound for N(σ,T). My thesis will improve their upper bound and also update the result to use better bounds on ζ on the half line among other improvements.