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Concept
On the Number of Primes Less Than a Given Magnitude
Summary
" die Anzahl der Primzahlen unter einer gegebenen " (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.
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 (ζ) for a function previously mentioned by Euler
The analytic continuation of this zeta function ζ(s) to all complex s ≠ 1
The entire function 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.
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