" die Anzahl der Primzahlen unter einer gegebenen " (usual

English English usually refers to: * English language * English people English may also refer to: Peoples, culture, and language * ''English'', an adjective for something of, from, or related to England ** English national ...

translation: "On the Number of Primes Less Than a Given Magnitude") is
seminal
9-page paper by

Bernhard Riemann Georg Friedrich Bernhard Riemann (; 17 September 1826 – 20 July 1866) was a German mathematician who made contributions to analysis, number theory, and differential geometry. In the field of real analysis, he is mostly known for the first ...

published in the November 1859 edition of the ''Monatsberichte der Königlich Preußischen Akademie der Wissenschaften zu Berlin''.

Overview

This paper studies the

prime-counting function In mathematics, the prime-counting function is the function counting the number of prime numbers less than or equal to some real number ''x''. It is denoted by (''x'') (unrelated to the number ). History Of great interest in number theory is ...

using analytic methods. Although it is the only paper Riemann ever published on

number theory Number theory (or arithmetic or higher arithmetic in older usage) is a branch of pure mathematics devoted primarily to the study of the integers and integer-valued functions. German mathematician Carl Friedrich Gauss (1777–1855) said, "Ma ...

, 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 A definition is a statement of the meaning of a term (a word, phrase, or other set of symbols). Definitions can be classified into two large categories: intensional definitions (which try to give the sense of a term), and extensional definiti ...

heuristic A heuristic (; ), or heuristic technique, is any approach to problem solving or self-discovery that employs a practical method that is not guaranteed to be optimal, perfect, or rational, but is nevertheless sufficient for reaching an immediate ...

arguments An argument is a statement or group of statements called premises intended to determine the degree of truth or acceptability of another statement called conclusion. Arguments can be studied from three main perspectives: the logical, the dialectic ...

, sketches of

proof Proof most often refers to: * Proof (truth), argument or sufficient evidence for the truth of a proposition * Alcohol proof, a measure of an alcoholic drink's strength Proof may also refer to: Mathematics and formal logic * Formal proof, a c ...

s, and the application of powerful analytic methods; all of these have become essential

concept Concepts are defined as abstract ideas. They are understood to be the fundamental building blocks of the concept behind principles, thoughts and beliefs. They play an important role in all aspects of cognition. As such, concepts are studied by ...

s and tools of

modern Modern may refer to: History *Modern history ** Early Modern period ** Late Modern period *** 18th century *** 19th century *** 20th century ** Contemporary history * Moderns, a faction of Freemasonry that existed in the 18th century Philosophy ...

analytic number theory In mathematics, analytic number theory is a branch of number theory that uses methods from mathematical analysis to solve problems about the integers. It is often said to have begun with Peter Gustav Lejeune Dirichlet's 1837 introduction of Diri ...

. Among the new definitions, ideas, and notation introduced: *The use of the

Greek letter The Greek alphabet has been used to write the Greek language since the late 9th or early 8th century BCE. It is derived from the earlier Phoenician alphabet, and was the earliest known alphabetic script to have distinct letters for vowels as ...

zeta Zeta (, ; uppercase Ζ, lowercase ζ; grc, ζῆτα, el, ζήτα, label= Demotic Greek, classical or ''zē̂ta''; ''zíta'') is the sixth letter of the Greek alphabet. In the system of Greek numerals, it has a value of 7. It was derived f ...

(ζ) for a function previously mentioned by

Euler Leonhard Euler ( , ; 15 April 170718 September 1783) was a Swiss mathematician, physicist, astronomer, geographer, logician and engineer who founded the studies of graph theory and topology and made pioneering and influential discoveries in ...

*The

analytic continuation In complex analysis, a branch of mathematics, analytic continuation is a technique to extend the domain of definition of a given analytic function. Analytic continuation often succeeds in defining further values of a function, for example in a ...

of this zeta function ζ(''s'') to all complex ''s'' ≠ 1 *The

entire function In complex analysis, an entire function, also called an integral function, is a complex-valued function that is holomorphic on the whole complex plane. Typical examples of entire functions are polynomials and the exponential function, and any fin ...

ξ(''s''), related to the zeta function through the

gamma function In mathematics, the gamma function (represented by , the capital letter gamma from the Greek alphabet) is one commonly used extension of the factorial function to complex numbers. The gamma function is defined for all complex numbers excep ...

(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 ''p''^''n''. (Riemann calls this function ''f''(''x'').) Among the proofs and sketches of proofs: *Two proofs of the

functional equation In mathematics, a functional equation is, in the broadest meaning, an equation in which one or several functions appear as unknowns. So, differential equations and integral equations are functional equations. However, a more restricted meaning ...

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 In mathematics, the Riemann hypothesis is the conjecture that the Riemann zeta function has its zeros only at the negative even integers and complex numbers with real part . Many consider it to be the most important unsolved problem in p ...

, 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, That is, (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: *Functional equations arising from

automorphic form In harmonic analysis and number theory, an automorphic form is a well-behaved function from a topological group ''G'' to the complex numbers (or complex vector space) which is invariant under the action of a discrete subgroup \Gamma \subset G o ...

s *

Analytic continuation In complex analysis, a branch of mathematics, analytic continuation is a technique to extend the domain of definition of a given analytic function. Analytic continuation often succeeds in defining further values of a function, for example in a ...

(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 found a formula for the

(''x'') (which he calls ''F''(''x'')). He notes that his equation explains the fact that (''x'') grows more slowly than the logarithmic integral, as had been found by

Carl Friedrich Gauss Johann Carl Friedrich Gauss (; german: Gauß ; la, Carolus Fridericus Gauss; 30 April 177723 February 1855) was a German mathematician and physicist who made significant contributions to many fields in mathematics and science. Sometimes refer ...

and

Carl Wolfgang Benjamin Goldschmidt Carl Wolfgang Benjamin Goldschmidt (4 August 1807 – 15 February 1851) was a German astronomer, mathematician, and physicist of Jewish descent who was a professor of astronomy at the University of Göttingen. He is also known as Benjamin Goldsch ...

. The paper contains some peculiarities for modern readers, such as the use of Π(''s'' − 1) instead of Γ(''s''), writing ''tt'' instead of ''t''², and using the bounds of ∞ to ∞ as to denote a

contour integral In the mathematical field of complex analysis, contour integration is a method of evaluating certain integrals along paths in the complex plane. Contour integration is closely related to the calculus of residues, a method of complex analysis. ...

References

*{{Citation , last=Edwards , first=H. M. , authorlink=Harold Edwards (mathematician) , year=1974 , title=Riemann's Zeta Function , publisher=Academic Press , location=New York , isbn=0-12-232750-0 , zbl=0315.10035

External links

Riemann's manuscriptUeber die Anzahl der Primzahlen unter einer gegebener Grösse
(transcription of Riemann's article)
On the Number of Primes Less Than a Given Magnitude
(English translation of Riemann's article) 1859 documents Analytic number theory Mathematics papers 1859 in science Works originally published in German magazines Works originally published in science and technology magazines Bernhard Riemann