Dirichlet theorem
A name referring to several theorems associated with Peter Gustav Lejeune Dirichlet (1805-1859).
Dirichlet's theorem in the theory of Diophantine approximations
For any real number $\alpha$ and any natural number $Q$ there exist integers $a$ and $q$ which satisfy the condition $$ |\alpha q - a | < \frac{1}{q}\,,\ \ \ 0 < q \le Q\ . $$ With the aid of the Dirichlet box principle a more general theorem can be demonstrated: For any real numbers $\alpha_1,\ldots,\alpha_n$ and any natural number $Q$ there exist integers $a_1,\ldots,a_n$ and $q$ such that $$ \max(|\alpha_1 q - a_1|,\ldots,|\alpha_n q - a_n|) < \frac{1}{Q^{1/n}}\,,\ \ \ 0 < q \le Q\ . $$
References
[1] | J.W.S. Cassels, "An introduction to diophantine approximation" , Cambridge Univ. Press (1957) |
V.I. Bernik
Dirichlet's unit theorem
A theorem describing the structure of the multiplicative group of units of an algebraic number field; obtained by P.G.L. Dirichlet [1].
Each algebraic number field of degree
over the field of rational numbers
has
different isomorphisms into the field of complex numbers
. If under the isomorphism
the image of the field is contained in the field of real numbers, this isomorphism is said to be real; otherwise it is said to be complex. Each complex isomorphism
has a complex conjugate isomorphism
, defined by the equation
,
. In this way the number
may be represented as
, where
is the number of real and
is the number of complex isomorphisms of
into
.
Dirichlet's theorem: In an arbitrary order of an algebraic number field
of degree
there exist
units
such that any unit
is uniquely representable as a product
![]() |
where are integers and
is some root of unity contained in
. The units
, the existence of which is established by Dirichlet's theorem, are said to be the basic units of the order
. In particular, the basic units of the maximal order
of the field
, i.e. the ring of integers of
, are usually called basic units of the algebraic number field
.
References
[1] | P.G.L. Dirichlet, "Werke" , 1 , Springer (1889) |
[2] | Z.I. Borevich, I.R. Shafarevich, "Number theory" , Acad. Press (1966) (Translated from Russian) (German translation: Birkhäuser, 1966) |
S.A. Stepanov
Dirichlet's theorem on prime numbers in an arithmetical progression
Each arithmetical progression whose first term and difference are relatively prime contains an infinite number of prime numbers. It was in fact proved by P.G.L. Dirichlet [1] that for any given relatively prime numbers ,
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where the summation is effected over all prime numbers subject to the condition
(
) and
is Euler's function. This relation may be interpreted as the law of uniform distribution of prime numbers over the residue classes
(
), since
![]() |
where the summation is extended over all prime numbers.
Let be an integer and let
be the amount of prime numbers
subject to the condition
(
), where
and
and
are relatively prime. Then
![]() |
where the estimate of the remainder is uniform in for any given
, and
is a magnitude which depends only on
(non-effectively). This is the modern form of Dirichlet's theorem, which immediately indicates the nature of the distribution of the prime numbers
(
) in the series of natural numbers. It is believed (the extended Riemann hypothesis) that, for given relatively prime
and
and any integer
,
![]() |
where is arbitrary, while
is a magnitude depending on
and
.
References
[1] | P.G.L. Dirichlet, "Vorlesungen über Zahlentheorie" , Vieweg (1894) |
[2] | K. Prachar, "Primzahlverteilung" , Springer (1957) |
[3] | A.A. Karatsuba, "Fundamentals of analytic number theory" , Moscow (1975) (In Russian) |
V.G. Sprindzhuk
Dirichlet's theorem on Fourier series
If a -periodic function
is piecewise monotone on the segment
and has at most finitely many discontinuity points on it, i.e. if the so-called Dirichlet conditions are satisfied, then its trigonometric Fourier series converges to
at each continuity point and to
at each discontinuity point. First demonstrated by P.G.L. Dirichlet [1]. Dirichlet's theorem was generalized by C. Jordan [3] to functions of bounded variation.
References
[1] | P.G.L. Dirichlet, "Sur la convergence des series trigonométriques qui servent à représenter une fonction arbitraire entre des limites donnés" J. Math. , 4 (1829) pp. 157–169 |
[2] | P.G.L. Dirichlet, "Werke" , 1 , Springer (1889) |
[3] | C. Jordan, C.R. Acad. Sci. , 92 (1881) pp. 228–230 |
[4] | N.K. [N.K. Bari] Bary, "A treatise on trigonometric series" , Pergamon (1964) (Translated from Russian) |
[5] | A. Zygmund, "Trigonometric series" , 1 , Cambridge Univ. Press (1988) |
Dirichlet theorem. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Dirichlet_theorem&oldid=44782