On the Number of Primes Less Than a Given Magnitude
Encyclopedia
die Anzahl der Primzahlen unter einer gegebenen (Usual English translation: On the Number of Primes Less Than a Given Magnitude) is a seminal 8-page paper by Bernhard Riemann
published in the November 1859 edition of the Monatsberichte der Königlich Preußischen Akademie der Wissenschaften zu Berlin.
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
. The paper was so influential that the notation s = σ + it is used to denote a complex number while discussing the zeta function (see below) instead of the usual z = x + iy. (The notation s = σ + it was begun by Edmund Landau
in 1903.)
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 Gauss
and a certain Goldschmidt
.
The paper contains some peculiarities for modern readers, such as the use of Π(s − 1) instead of Γ(s), or writing tt instead of t2. The style can also be surprising, such as writing an integral from ∞ to ∞.
Bernhard Riemann
Georg Friedrich Bernhard Riemann was an influential German mathematician who made lasting contributions to analysis and differential geometry, some of them enabling the later development of general relativity....
published in the November 1859 edition of the Monatsberichte der Königlich Preußischen Akademie der Wissenschaften zu Berlin.
Although it is the only paper he ever published on number theory
Number theory
Number theory is a branch of pure mathematics devoted primarily to the study of the integers. Number theorists study prime numbers as well...
, 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
Sketch (drawing)
A sketch is a rapidly executed freehand drawing that is not usually intended as a finished work...
of proofs, and the application of powerful analytic methods; all of these have become essential concepts and tools of modern analytic number theory
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 Dirichlet's introduction of Dirichlet L-functions to give the first proof of Dirichlet's theorem on arithmetic...
. The paper was so influential that the notation s = σ + it is used to denote a complex number while discussing the zeta function (see below) instead of the usual z = x + iy. (The notation s = σ + it was begun by Edmund Landau
Edmund Landau
Edmund Georg Hermann Landau was a German Jewish mathematician who worked in the fields of number theory and complex analysis.-Biography:...
in 1903.)
Among the new definitions, ideas, and notation introduced:
- The use of the GreekGreek alphabetThe Greek alphabet is the script that has been used to write the Greek language since at least 730 BC . The alphabet in its classical and modern form consists of 24 letters ordered in sequence from alpha to omega...
letterLetter (alphabet)A letter is a grapheme in an alphabetic system of writing, such as the Greek alphabet and its descendants. Letters compose phonemes and each phoneme represents a phone in the spoken form of the language....
zetaZeta-Science:* Zeta functions, in mathematics** Riemann zeta function* Zeta potential, the electrokinetic potential of a colloidal system* Tropical Storm Zeta , formed in December 2005 and lasting through January 2006* Z-pinch, in fusion power...
(ζ) for a functionFunction (mathematics)In mathematics, a function associates one quantity, the argument of the function, also known as the input, with another quantity, the value of the function, also known as the output. A function assigns exactly one output to each input. The argument and the value may be real numbers, but they can...
previously mentioned by Euler - The analytic continuationAnalytic continuationIn complex analysis, a branch of mathematics, analytic continuation is a technique to extend the domain of a given analytic function. Analytic continuation often succeeds in defining further values of a function, for example in a new region where an infinite series representation in terms of which...
of this zeta function ζ(s) to all complexComplex numberA complex number is a number consisting of a real part and an imaginary part. Complex numbers extend the idea of the one-dimensional number line to the two-dimensional complex plane by using the number line for the real part and adding a vertical axis to plot the imaginary part...
s ≠ 1 - The entire functionEntire functionIn complex analysis, an entire function, also called an integral function, is a complex-valued function that is holomorphic over the whole complex plane...
ξ(s), related to the zeta function through the gamma functionGamma functionIn mathematics, the gamma function is an extension of the factorial function, with its argument shifted down by 1, to real and complex numbers...
(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 pn. (Riemann calls this function f(x).)
Among the proofs and sketches of proofs:
- 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 parts lie between 0 and T.
Among the conjectures made:
- The Riemann hypothesisRiemann hypothesisIn mathematics, the Riemann hypothesis, proposed by , is a conjecture about the location of the zeros of the Riemann zeta function which states that all non-trivial zeros have real part 1/2...
, 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, "… es ist sehr wahrscheinlich, dass alle Wurzeln reell sind. 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." That is, "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."(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 continuationAnalytic continuationIn complex analysis, a branch of mathematics, analytic continuation is a technique to extend the domain of a given analytic function. Analytic continuation often succeeds in defining further values of a function, for example in a new region where an infinite series representation in terms of which...
(although not in the spirit of Weierstrass) - Contour integrationMethods of contour integrationIn 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 methodology of complex analysis....
- 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
Lebesgue-Stieltjes integration
In measure-theoretic analysis and related branches of mathematics, Lebesgue–Stieltjes integration generalizes Riemann–Stieltjes and Lebesgue integration, preserving the many advantages of the former in a more general measure-theoretic framework...
. 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 Gauss
Carl Friedrich Gauss
Johann Carl Friedrich Gauss was a German mathematician and scientist who contributed significantly to many fields, including number theory, statistics, analysis, differential geometry, geodesy, geophysics, electrostatics, astronomy and optics.Sometimes referred to as the Princeps mathematicorum...
and a certain Goldschmidt
Goldschmidt
Goldschmidt is a German surname meaning "Goldsmith". It may refer to:* Adolph Goldschmidt , art historian* Berthold Goldschmidt , composer* Hans Goldschmidt , chemist, son of Theodor Goldschmidt...
.
The paper contains some peculiarities for modern readers, such as the use of Π(s − 1) instead of Γ(s), or writing tt instead of t2. The style can also be surprising, such as writing an integral from ∞ to ∞.