An introduction to the Riemann hypothesis

Abstract: This paper exhibits the intertwinement between the prime numbers and the zeros of the Riemann zeta function, drawing upon existing literature by Davenport, Ahlfors, et al. We begin with the meromorphic continuation of the Riemann zeta function ζ and the gamma function Γ . We then derive a functional equation that relates these functions and formulate the Riemann hypothesis. We move on to the topic of nite-ordered functions and their Hadamard products. We show that the xi function ξ is of finite order, whence we obtain many useful properties. We then use these properties to and a zero-free region for ζ in the critical strip. We also determine the vertical distribution of the non-trivial zeros. We finally use Perron's formula to derive von Mangoldt's explicit formula, which is an approximation of the Chebyshevfunction ψ . Using this approximation, we prove the prime number theorem and conclude with an implication ofthe Riemann hypothesis.