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

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.