" 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 Georg Friedrich Bernhard Riemann (; ; 17September 182620July 1866) was a German mathematician who made profound contributions to analysis, number theory, and differential geometry. In the field of real analysis, he is mostly known for the f ...

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 . It is denoted by (unrelated to the number ). A symmetric variant seen sometimes is , which is equal ...

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

number theory Number theory is a branch of pure mathematics devoted primarily to the study of the integers and arithmetic functions. Number theorists study prime numbers as well as the properties of mathematical objects constructed from integers (for example ...

, 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 definit ...

heuristic A heuristic or heuristic technique (''problem solving'', '' mental shortcut'', ''rule of thumb'') is any approach to problem solving that employs a pragmatic method that is not fully optimized, perfected, or rationalized, but is nevertheless ...

arguments An argument is a series of sentences, statements, or propositions some of which are called premises and one is the conclusion. The purpose of an argument is to give reasons for one's conclusion via justification, explanation, and/or persua ...

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

concept A concept is an abstract idea that serves as a foundation for more concrete principles, thoughts, and beliefs. Concepts play an important role in all aspects of cognition. As such, concepts are studied within such disciplines as linguistics, ...

s and tools of modern

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 Dir ...

. 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 BC. It was derived from the earlier Phoenician alphabet, and is the earliest known alphabetic script to systematically write vowels as wel ...

zeta Zeta (, ; uppercase Ζ, lowercase ζ; , , 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 from the Phoenician alphabet, Phoenician letter zay ...

(ζ) for a function previously mentioned by Euler *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 ne ...

of this zeta function ζ(''s'') to all

complex Complex commonly refers to: * Complexity, the behaviour of a system whose components interact in multiple ways so possible interactions are difficult to describe ** Complex system, a system composed of many components which may interact with each ...

''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 ...

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

gamma function In mathematics, the gamma function (represented by Γ, capital Greek alphabet, Greek letter gamma) is the most common extension of the factorial function to complex numbers. Derived by Daniel Bernoulli, the gamma function \Gamma(z) is defined ...

(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 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 pure ...

, 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 forms *

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 ne ...

(although not in the spirit of Weierstrass) *

Contour integration 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 Residue theorem, calculus of residues, a method of co ...

* 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 (; ; ; 30 April 177723 February 1855) was a German mathematician, astronomer, geodesist, and physicist, who contributed to many fields in mathematics and science. He was director of the Göttingen Observatory and ...

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 ''t''², and using the bounds of ∞ to ∞ as to denote a contour integral.

References

External links

Riemann's manuscriptUeber die Anzahl der Primzahlen unter einer gegebener Grösse
(transcription of Riemann's article)
On the Number of Prime Numbers less than a Given Quantity
(English translation of Riemann's article) {{Bernhard Riemann 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