On the Number of Primes Less Than a Given Magnitude

The article

"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 10-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:

Among the proofs and sketches of proofs:

Among the conjectures made:

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. He then obtained the main result of the paper, a formula for J(x), by comparing with ln(ζ(s)). Riemann then found a formula for the prime-counting function π(x) (which he calls F(x)). He notes that his equation explains the fact that π(x) 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 Π(s  1) instead of Γ(s), writing tt instead of t2, and using the bounds of ∞ to ∞ as to denote a contour integral.

References

External links

This article is issued from Wikipedia - version of the Tuesday, March 22, 2016. The text is available under the Creative Commons Attribution/Share Alike but additional terms may apply for the media files.