On Riemann's Paper, "On the Number of Primes Less Than a Given Magnitude"

TL;DR

This paper provides a detailed analysis of Riemann's 1859 work on the distribution of prime numbers, highlighting his novel concepts like analytical continuation, the product formula, and the zeros of the zeta function, which underpin modern prime number theory.

Contribution

It offers an in-depth examination of Riemann's original article, emphasizing the introduction of key mathematical concepts that revolutionized understanding of prime distribution.

Findings

Analysis of Riemann's use of analytical continuation

Discussion of the product formula for entire functions

Study of zeros of the Riemann zeta function

Abstract

This paper is devoted to one of the members of the G\"ottingen triumvirate, Gau{\ss}, Dirichlet and Riemann. It is the latter to whom I wish to pay tribute, and especially to his world-famous article of 1859, which he presented in person at the Berlin Academy upon his election as a corresponding member. His article, entitled, "Uber die Anzahl der Primzahlen unter einer gegebenen Gr\"o{\ss}e" ("On the Number of Primes Less Than a Given Magnitude"), revolutionized mathematics worldwide. Included in the present paper is a detailed analysis of Riemann's article, including such novel concepts as analytical continuation in the complex plane; the product formula for entire functions; and, last but not least, a detailed study of the zeros of the so-called Riemann zeta function and its close relation to determining the number of primes up to a given magnitude, i.e., an explicit formula for the prime counting function.