On the Number of Primes Less Than a Given Magnitude

" die Anzahl der Primzahlen unter einer gegebenen " is a seminal 9-page paper by Bernhard Riemann published in the November 1859 edition of the Monatsberichte der Königlich Preußischen Akademie der Wissenschaften zu Berlin.

Overview

This paper studies the prime-counting function using analytic methods. Although it is the only paper Riemann ever published on number theory, it contains ideas which influenced thousands of researchers during the late 19th century and up to the present day. The paper consists primarily of definitions, heuristic arguments, sketches of proofs, and the application of powerful analytic methods; all of these have become essential concepts and tools of modern analytic number theory.
Among the new definitions, ideas, and notation introduced:

The use of the Greek letter zeta for a function previously mentioned by Euler
The analytic continuation of this [Riemann zeta function|zeta Function (mathematics)|function ζ(s)] to all complex
The entire function ξ, related to the zeta function through the gamma function
The discrete function J defined for, which is defined by and J jumps by 1/n at each prime power pⁿ.

Among the proofs and sketches of proofs:

Two proofs of the functional equation of ζ
Proof sketch of the product representation of ξ
Proof sketch of the approximation of the number of roots of ξ whose imaginary parts lie between 0 and T.

Among the conjectures made:

The Riemann hypothesis, that all zeros of ζ have real part 1/2. Riemann states this in terms of the roots of the related ξ function, That is,

New methods and techniques used in number theory:

Functional equations arising from automorphic forms
Analytic continuation
Contour integration
Fourier inversion.

Riemann also discussed the relationship between ζ and the distribution of the prime numbers, using the function J essentially as a measure for Stieltjes integration. He then obtained the main result of the paper, a formula for J, by comparing with ln. Riemann then found a formula for the prime-counting function π. He notes that his equation explains the fact that π grows more slowly than the logarithmic integral, as had been found by Carl Friedrich Gauss and Carl Wolfgang Benjamin Goldschmidt.
The paper contains some peculiarities for modern readers, such as the use of instead of Γ, writing tt instead of t², and using the bounds of ∞ to ∞ as to denote a contour integral.