Riemann’s Zeta and Two Arithmetic Functions

Abstract

An intriguing connection exists between the Riemann zeta-function (s) and both the von Mangoldt function (n) and the M obius function (n): With appropriate information about (s) this connection provides a door to nding bounds for the sums (x) ?? x and M(x) that involve these arithmetic functions (n) and (n): A bound for (x)??x bounds the error in the well-known Prime Number Theorem. In this thesis we consider a particular approach to nding bounds that is presented in Problems in Analytic Number Theory by Murty. Using this approach we prove implicit bounds for (x)??x and M(x) after providing background on the zeta-function, presenting a bird's-eye view of the approach and exploring the history of bounding these sums. The implicit bounds we prove are by no means new. However, we did not come across any explicit form of the bound we prove for M(x) in the literature. Time constraints prevented the completion of such an explicit bound during the course of this thesis, but we do include some details relevant to such a task. Show more