On the Number of Primes Less Than a Given Magnitude

"Ueber die Anzahl der Primzahlen unter einer gegebenen Grösse" (usual English translation: "On the Number of Primes Less Than a Given Magnitude") 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

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

Euler

The analytic continuation of this zeta function ζ(s) to all complex s ≠ 1
The
ξ(s), related to the zeta function through the gamma function (or the Π function
, in Riemann's usage)

The discrete function J(x) defined for x ≥ 0, which is defined by J(0) = 0 and J(x) jumps by 1/n at each prime power pⁿ. (Riemann calls this function f(x).)

Among the proofs and sketches of proofs:

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

Among the conjectures made:

The Riemann hypothesis, that all (nontrivial) zeros of ζ(s) have real part 1/2. Riemann states this in terms of the roots of the related ξ function,
... es ist sehr wahrscheinlich, dass alle Wurzeln reell sind. Hiervon wäre allerdings ein strenger Beweis zu wünschen; ich habe indess die Aufsuchung desselben nach einigen flüchtigen vergeblichen Versuchen vorläufig bei Seite gelassen, da er für den nächsten Zweck meiner Untersuchung entbehrlich schien.
That is,
it is very probable that all roots are real. One would, however, wish for a strict proof of this; I have, though, after some fleeting futile attempts, provisionally put aside the search for such, as it appears unnecessary for the next objective of my investigation.
(He was discussing a version of the zeta function, modified so that its roots are real rather than on the critical line.)

New methods and techniques used in number theory: