The Riemann Hypothesis

Abstract: The Riemann hypothesis was first proposed by Bernhard Riemann in 1860 [1] and says all non-trivial zeroes to the Riemann zeta function lie on the line with the real part 12 in the complex plane [1]. If proven to be true this would give a much better approximation of the number of prime numbers less than some number X. The Riemann hypothesis is regarded to be one of the most important unsolved mathematical problems. It is one of the Clay InstituteMilleniumproblems and originally one of the unsolved problems presented by David Hilbert as essential for 20th century mathematics at International Congress ofMathematics in 1900. It is the aim of this report to illustrate how the zeros to the zeta function affects the approximation of the number of primes less than X. We will start out by defining some core concepts in chapters 2,3 and 4 and then move on to some theory about integral functions of order 1. This theory will then be applied a function of interest. We then move on and use the results to derive the prime number theorem.