Notes on the Riemann Hypothesis

TL;DR

This paper provides an introductory overview of the Riemann Hypothesis, exploring its historical context, mathematical background, and key developments in the study of zeta and L-functions since Riemann's time.

Contribution

It offers a comprehensive survey of the Riemann Hypothesis, combining historical insights with recent progress in understanding zeta and L-functions.

Findings

Historical analysis of Riemann's original work

Overview of major developments in the field

Discussion of current understanding and open problems

Abstract

These notes were written from a series of lectures given in March 2010 at the Universidad Complutense of Madrid and then in Barcelona for the centennial anniversary of the Spanish Mathematical Society (RSME). Our aim is to give an introduction to the Riemann Hypothesis and a panoramic view of the world of zeta and L-functions. We first review Riemann's foundational article and discuss the mathematical background of the time and his possible motivations for making his famous conjecture. We discuss some of the most relevant developments after Riemann that have contributed to a better understanding of the conjecture.