On the Number of Primes Less Than a Given Magnitude (or Über die Anzahl der Primzahlen unter einer gegebenen Grösse) is a seminal 8-page paper by Bernhard Riemann published in the November 1859 edition of the Monthly Reports of the Berlin Academy. Although it is the only paper he 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. Bernhard Riemann. ... 1859 is a common year starting on Saturday. ... Traditionally, number theory is that branch of pure mathematics concerned with the properties of integers. ... Alternative meaning: Nineteenth Century (periodical) (18th century — 19th century — 20th century — more centuries) As a means of recording the passage of time, the 19th century was that century which lasted from 1801-1900 in the sense of the Gregorian calendar. ... Analytic number theory is the branch of number theory that uses methods from mathematical analysis. ...

Among the new definitions introduced:

• The analytic continuation of the Riemann zeta function ζ(s) to all complex s ≠ 1
• The entire function ξ(s)
• 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

Among the proofs and sketches of proofs: In complex analysis, a branch of mathematics, analytic continuation is a technique to extend the domain of definition of a given analytic function. ... In mathematics, the Riemann zeta function is a function which is of paramount importance in number theory, because of its relation to the distribution of prime numbers. ... The complex numbers are an extension of the real numbers, in which all non-constant polynomials have roots. ... In complex analysis, an entire function is a function that is holomorphic everywhere on the whole complex plane. ...

• Two proofs of the functional equation of ζ(s)
• "Proof" of the product representation of ξ(s)
• "Proof" of the approximation of the number of roots of ξ(s) whose imaginary part lies between 0 and T

Among conjectures made: In mathematics, the L-functions of number theory have certain functional equations, as one of their characteristic properties. ...

• The Riemann hypothesis, that all zeros of ζ(s) have real part 1/2

New methods and techniques used in number theory: In mathematics, the Riemann hypothesis (aka Riemann zeta hypothesis), first formulated by Bernhard Riemann in 1859, is one of the most famous of all unsolved problems. ...

• Analytic continuation (although not in the spirit of Weierstrass)
• Contour integration
• Fourier inversion

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 attempted to make an approximate formula for the prime-counting function π(x), although he himself admits he is aware of the defects of his arguments. In complex analysis, a branch of mathematics, analytic continuation is a technique to extend the domain of definition of a given analytic function. ... This article is about path integrals in the general mathematical sense, and not the path integral formulation of physics which was studied by Richard Feynman. ... In mathematics, Fourier inversion recovers a function from its Fourier transform. ... If you are having difficulty understanding this article, you might want to first learn more about integrals, particularly the Lebesgue integral, and measure theory. ...

External links

• English translation of Riemann's paper (http://www.maths.tcd.ie/pub/HistMath/People/Riemann/Zeta/EZeta.pdf)
• Number theory and physics website (http://www.maths.ex.ac.uk/~mwatkins)

References

• Riemann's Zeta Function, H. M. Edwards, Dover, 1974, ISBN 0486417409