On the Number of Primes Less Than a Given Magnitude

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.

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.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.New methods and techniques used in number theory: Riemann also discussed the relationship between ζ(s) and the distribution of the prime numbers, using the function J(x) essentially as a measure for Stieltjes integration.

The paper contains some peculiarities for modern readers, such as the use of Π(s − 1) instead of Γ(s), writing tt instead of t2, and using the bounds of ∞ to ∞ as to denote a contour integral.