The Riemann Zeta Function

Abstract

The Riemann Zeta function is an enigma for mathematicians. What is more famously known is the Riemann Hypothesis, which was introduced to the world formally by David Hilbert in 1900 as one of his 23 unsolved problems. This problem has gone unsolved for 163 years and counting.The zeta function was first introduced by Leonhard Euler in the mid 18th century where it was defined over the real numbers. Euler then makes a monumental connection between the zeta function and the product of prime numbers, here the zeta function gains its infamy. Bernhard Riemann comes along in 1859 and extends the zeta function to the complex numbers. In doing so, he introduces his famous hypothesis ”The non trivial zeros of the function have a real part of 1/2 ”. This hypothesis implies that we can gain insight into the distribution of the prime numbers. In mathematics, we can calculate how many prime numbers there are in a defined interval, but we don’t know how to locate them.We intend to look at the Riemann Zeta function through two lenses, graphically and analytically. We explore how this function behaves under certain conditions and assumptions, while also being able to make meaningful visualizations that helps us understand the behavior of this function. We will also examine the analytic continuation of the Riemann Zeta function, which is the key element that allows the Riemann Hypothesis to exist in the first place. We will also look at the progress being made to solve this hypothesis.