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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
+
{{TEX|done}}
 +
 
 +
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\ .
 
|\alpha q - a | < \frac{1}{q}\,,\ \ \ 0 < q \le Q\ .
Line 15: Line 20:
 
''V.I. Bernik''
 
''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 [[#References|[1]]].
+
===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 [[#References|[1]]].
  
Each algebraic number field <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d032/d032940/d03294011.png" /> of degree <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d032/d032940/d03294012.png" /> over the field of rational numbers <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d032/d032940/d03294013.png" /> has <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d032/d032940/d03294014.png" /> different isomorphisms into the field of complex numbers <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d032/d032940/d03294015.png" />. If under the isomorphism <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d032/d032940/d03294016.png" /> 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 <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d032/d032940/d03294017.png" /> has a complex conjugate isomorphism <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d032/d032940/d03294018.png" />, defined by the equation <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d032/d032940/d03294019.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d032/d032940/d03294020.png" />. In this way the number <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d032/d032940/d03294021.png" /> may be represented as <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d032/d032940/d03294022.png" />, where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d032/d032940/d03294023.png" /> is the number of real and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d032/d032940/d03294024.png" /> is the number of complex isomorphisms of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d032/d032940/d03294025.png" /> into <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d032/d032940/d03294026.png" />.
+
Each algebraic number field $  K $
 +
of degree $  n $
 +
over the field of rational numbers $  \mathbf Q $
 +
has $  n $
 +
different isomorphisms into the field of complex numbers $  \mathbf C $.  
 +
If under the isomorphism $  \sigma : K \rightarrow \mathbf C $
 +
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 $  \sigma $
 +
has a complex conjugate isomorphism $  \overline \sigma \; : K \rightarrow \mathbf C $,  
 +
defined by the equation $  \overline \sigma \; ( \alpha ) = \overline{ {\sigma ( \alpha ) }}\; $,  
 +
$  \alpha \in K $.  
 +
In this way the number $  n $
 +
may be represented as $  n = s + 2t $,  
 +
where $  s $
 +
is the number of real and $  2t $
 +
is the number of complex isomorphisms of $  K $
 +
into $  \mathbf C $.
  
Dirichlet's theorem: In an arbitrary [[Order|order]] <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d032/d032940/d03294027.png" /> of an algebraic number field <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d032/d032940/d03294028.png" /> of degree <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d032/d032940/d03294029.png" /> there exist <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d032/d032940/d03294030.png" /> units <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d032/d032940/d03294031.png" /> such that any unit <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d032/d032940/d03294032.png" /> is uniquely representable as a product
+
Dirichlet's theorem: In an arbitrary [[Order|order]] $  A $
 +
of an algebraic number field $  K $
 +
of degree $  n = s + 2t $
 +
there exist $  r = s + t - 1 $
 +
units $  \epsilon _ {1} \dots \epsilon _ {r} $
 +
such that any unit $  \epsilon \in A $
 +
is uniquely representable as a product
  
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d032/d032940/d03294033.png" /></td> </tr></table>
+
$$
 +
\epsilon  = \zeta \epsilon _ {1} ^ {s _ {1} } \dots \epsilon _ {r} ^ {s _ {r} } ,
 +
$$
  
where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d032/d032940/d03294034.png" /> are integers and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d032/d032940/d03294035.png" /> is some root of unity contained in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d032/d032940/d03294036.png" />. The units <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d032/d032940/d03294037.png" />, the existence of which is established by Dirichlet's theorem, are said to be the basic units of the order <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d032/d032940/d03294038.png" />. In particular, the basic units of the maximal order <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d032/d032940/d03294039.png" /> of the field <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d032/d032940/d03294040.png" />, i.e. the ring of integers of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d032/d032940/d03294041.png" />, are usually called basic units of the algebraic number field <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d032/d032940/d03294042.png" />.
+
where $  s _ {1} \dots s _ {r} $
 +
are integers and $  \zeta $
 +
is some root of unity contained in $  A $.  
 +
The units $  \epsilon _ {1} \dots \epsilon _ {r} $,  
 +
the existence of which is established by Dirichlet's theorem, are said to be the basic units of the order $  A $.  
 +
In particular, the basic units of the maximal order $  D $
 +
of the field $  K $,  
 +
i.e. the ring of integers of $  K $,  
 +
are usually called basic units of the algebraic number field $  K $.
  
 
====References====
 
====References====
Line 30: Line 67:
 
''S.A. Stepanov''
 
''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 [[#References|[1]]] that for any given relatively prime numbers <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d032/d032940/d03294043.png" />,
+
===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 [[#References|[1]]] that for any given relatively prime numbers $  l , k $,
 +
 
 +
$$
 +
\lim\limits _ {s \rightarrow 1 + 0 }  \sum _ { p }
 +
 
 +
\frac{1}{p  ^ {s} }
 +
 +
\frac{1}{ \mathop{\rm ln}  1 / ( s - 1 ) }
 +
  =
 +
\frac{1}{\phi (k) }
 +
,
 +
$$
  
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d032/d032940/d03294044.png" /></td> </tr></table>
+
where the summation is effected over all prime numbers  $  p $
 +
subject to the condition  $  p \equiv l $(
 +
$  \mathop{\rm mod}  k $)
 +
and  $  \phi (k) $
 +
is Euler's function. This relation may be interpreted as the law of uniform distribution of prime numbers over the residue classes  $  l $(
 +
$  \mathop{\rm mod}  k $),
 +
since
  
where the summation is effected over all prime numbers <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d032/d032940/d03294045.png" /> subject to the condition <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d032/d032940/d03294046.png" /> (<img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d032/d032940/d03294047.png" />) and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d032/d032940/d03294048.png" /> is Euler's function. This relation may be interpreted as the law of uniform distribution of prime numbers over the residue classes <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d032/d032940/d03294050.png" /> (<img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d032/d032940/d03294051.png" />), since
+
$$
 +
\lim\limits _ {s \rightarrow 1 + 0 }  \sum _ { p }
  
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d032/d032940/d03294052.png" /></td> </tr></table>
+
\frac{1}{p  ^ {s} }
 +
 +
\frac{1}{ \mathop{\rm ln}  1 / ( s - 1 ) }
 +
  = 1 ,
 +
$$
  
 
where the summation is extended over all prime numbers.
 
where the summation is extended over all prime numbers.
  
Let <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d032/d032940/d03294053.png" /> be an integer and let <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d032/d032940/d03294054.png" /> be the amount of prime numbers <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d032/d032940/d03294055.png" /> subject to the condition <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d032/d032940/d03294056.png" /> (<img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d032/d032940/d03294057.png" />), where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d032/d032940/d03294058.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d032/d032940/d03294059.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d032/d032940/d03294060.png" /> are relatively prime. Then
+
Let $  x > 1 $
 +
be an integer and let $  \pi ( x ;  l , k ) $
 +
be the amount of prime numbers $  p \leq  x $
 +
subject to the condition $  p \equiv l $(
 +
$  \mathop{\rm mod}  k $),  
 +
where $  0 < l < k $
 +
and $  l $
 +
and $  k $
 +
are relatively prime. Then
  
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d032/d032940/d03294061.png" /></td> </tr></table>
+
$$
 +
\pi ( x ; l , k )  =
 +
\frac{\int\limits _ { 2 } ^ { x } 
 +
\frac{d u }{ \mathop{\rm ln}
 +
u }
 +
}{\phi (k) }
 +
+ O ( x e ^ {-c \sqrt { \mathop{\rm ln}  x } } ) ,
 +
$$
  
where the estimate of the remainder is uniform in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d032/d032940/d03294062.png" /> for any given <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d032/d032940/d03294063.png" />, and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d032/d032940/d03294064.png" /> is a magnitude which depends only on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d032/d032940/d03294065.png" /> (non-effectively). This is the modern form of Dirichlet's theorem, which immediately indicates the nature of the distribution of the prime numbers <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d032/d032940/d03294066.png" /> (<img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d032/d032940/d03294067.png" />) in the series of natural numbers. It is believed (the extended Riemann hypothesis) that, for given relatively prime <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d032/d032940/d03294068.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d032/d032940/d03294069.png" /> and any integer <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d032/d032940/d03294070.png" />,
+
where the estimate of the remainder is uniform in $  k \leq  (  \mathop{\rm ln}  x )  ^ {A} $
 +
for any given $  A > 0 $,  
 +
and $  c = c (A) > 0 $
 +
is a magnitude which depends only on $  A $(
 +
non-effectively). This is the modern form of Dirichlet's theorem, which immediately indicates the nature of the distribution of the prime numbers $  p \equiv l $(
 +
$  \mathop{\rm mod}  k $)  
 +
in the series of natural numbers. It is believed (the extended Riemann hypothesis) that, for given relatively prime $  l $
 +
and $  k $
 +
and any integer $  x > 1 $,
  
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d032/d032940/d03294071.png" /></td> </tr></table>
+
$$
 +
\pi ( x ; l , k )  =
 +
\frac{\int\limits _ { 2 } ^ { x } 
 +
\frac{d u }{ \mathop{\rm ln} \
 +
u }
 +
}{\phi (k) }
 +
+ O ( x ^ {1 / 2 + \epsilon } ) ,
 +
$$
  
where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d032/d032940/d03294072.png" /> is arbitrary, while <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d032/d032940/d03294073.png" /> is a magnitude depending on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d032/d032940/d03294074.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d032/d032940/d03294075.png" />.
+
where $  \epsilon > 0 $
 +
is arbitrary, while $  O $
 +
is a magnitude depending on $  k $
 +
and $  \epsilon $.
  
 
====References====
 
====References====
Line 55: Line 148:
 
''V.G. Sprindzhuk''
 
''V.G. Sprindzhuk''
  
Dirichlet's theorem on Fourier series: If a <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d032/d032940/d03294076.png" />-periodic function <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d032/d032940/d03294077.png" /> is piecewise monotone on the segment <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d032/d032940/d03294078.png" /> 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 <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d032/d032940/d03294079.png" /> at each continuity point and to <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d032/d032940/d03294080.png" /> at each discontinuity point. First demonstrated by P.G.L. Dirichlet [[#References|[1]]]. Dirichlet's theorem was generalized by C. Jordan [[#References|[3]]] to functions of bounded variation.
+
===Dirichlet's theorem on Fourier series===
 +
If a  $  2 \pi $-
 +
periodic function $  f $
 +
is piecewise monotone on the segment $  [ - \pi , \pi ] $
 +
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 $  f (x) $
 +
at each continuity point and to $  [ f ( x + 0 ) + f ( x - 0 ) ]/ 2 $
 +
at each discontinuity point. First demonstrated by P.G.L. Dirichlet [[#References|[1]]]. Dirichlet's theorem was generalized by C. Jordan [[#References|[3]]] to functions of bounded variation.
  
 
====References====
 
====References====
 
<table><TR><TD valign="top">[1]</TD> <TD valign="top">  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</TD></TR><TR><TD valign="top">[2]</TD> <TD valign="top">  P.G.L. Dirichlet,  "Werke" , '''1''' , Springer  (1889)</TD></TR><TR><TD valign="top">[3]</TD> <TD valign="top">  C. Jordan,  ''C.R. Acad. Sci.'' , '''92'''  (1881)  pp. 228–230</TD></TR><TR><TD valign="top">[4]</TD> <TD valign="top">  N.K. [N.K. Bari] Bary,  "A treatise on trigonometric series" , Pergamon  (1964)  (Translated from Russian)</TD></TR><TR><TD valign="top">[5]</TD> <TD valign="top">  A. Zygmund,  "Trigonometric series" , '''1''' , Cambridge Univ. Press  (1988)</TD></TR></table>
 
<table><TR><TD valign="top">[1]</TD> <TD valign="top">  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</TD></TR><TR><TD valign="top">[2]</TD> <TD valign="top">  P.G.L. Dirichlet,  "Werke" , '''1''' , Springer  (1889)</TD></TR><TR><TD valign="top">[3]</TD> <TD valign="top">  C. Jordan,  ''C.R. Acad. Sci.'' , '''92'''  (1881)  pp. 228–230</TD></TR><TR><TD valign="top">[4]</TD> <TD valign="top">  N.K. [N.K. Bari] Bary,  "A treatise on trigonometric series" , Pergamon  (1964)  (Translated from Russian)</TD></TR><TR><TD valign="top">[5]</TD> <TD valign="top">  A. Zygmund,  "Trigonometric series" , '''1''' , Cambridge Univ. Press  (1988)</TD></TR></table>
 
{{TEX|part}}
 

Latest revision as of 14:05, 17 March 2020


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 $ K $ of degree $ n $ over the field of rational numbers $ \mathbf Q $ has $ n $ different isomorphisms into the field of complex numbers $ \mathbf C $. If under the isomorphism $ \sigma : K \rightarrow \mathbf C $ 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 $ \sigma $ has a complex conjugate isomorphism $ \overline \sigma \; : K \rightarrow \mathbf C $, defined by the equation $ \overline \sigma \; ( \alpha ) = \overline{ {\sigma ( \alpha ) }}\; $, $ \alpha \in K $. In this way the number $ n $ may be represented as $ n = s + 2t $, where $ s $ is the number of real and $ 2t $ is the number of complex isomorphisms of $ K $ into $ \mathbf C $.

Dirichlet's theorem: In an arbitrary order $ A $ of an algebraic number field $ K $ of degree $ n = s + 2t $ there exist $ r = s + t - 1 $ units $ \epsilon _ {1} \dots \epsilon _ {r} $ such that any unit $ \epsilon \in A $ is uniquely representable as a product

$$ \epsilon = \zeta \epsilon _ {1} ^ {s _ {1} } \dots \epsilon _ {r} ^ {s _ {r} } , $$

where $ s _ {1} \dots s _ {r} $ are integers and $ \zeta $ is some root of unity contained in $ A $. The units $ \epsilon _ {1} \dots \epsilon _ {r} $, the existence of which is established by Dirichlet's theorem, are said to be the basic units of the order $ A $. In particular, the basic units of the maximal order $ D $ of the field $ K $, i.e. the ring of integers of $ K $, are usually called basic units of the algebraic number field $ K $.

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 $ l , k $,

$$ \lim\limits _ {s \rightarrow 1 + 0 } \sum _ { p } \frac{1}{p ^ {s} } \frac{1}{ \mathop{\rm ln} 1 / ( s - 1 ) } = \frac{1}{\phi (k) } , $$

where the summation is effected over all prime numbers $ p $ subject to the condition $ p \equiv l $( $ \mathop{\rm mod} k $) and $ \phi (k) $ is Euler's function. This relation may be interpreted as the law of uniform distribution of prime numbers over the residue classes $ l $( $ \mathop{\rm mod} k $), since

$$ \lim\limits _ {s \rightarrow 1 + 0 } \sum _ { p } \frac{1}{p ^ {s} } \frac{1}{ \mathop{\rm ln} 1 / ( s - 1 ) } = 1 , $$

where the summation is extended over all prime numbers.

Let $ x > 1 $ be an integer and let $ \pi ( x ; l , k ) $ be the amount of prime numbers $ p \leq x $ subject to the condition $ p \equiv l $( $ \mathop{\rm mod} k $), where $ 0 < l < k $ and $ l $ and $ k $ are relatively prime. Then

$$ \pi ( x ; l , k ) = \frac{\int\limits _ { 2 } ^ { x } \frac{d u }{ \mathop{\rm ln} u } }{\phi (k) } + O ( x e ^ {-c \sqrt { \mathop{\rm ln} x } } ) , $$

where the estimate of the remainder is uniform in $ k \leq ( \mathop{\rm ln} x ) ^ {A} $ for any given $ A > 0 $, and $ c = c (A) > 0 $ is a magnitude which depends only on $ A $( non-effectively). This is the modern form of Dirichlet's theorem, which immediately indicates the nature of the distribution of the prime numbers $ p \equiv l $( $ \mathop{\rm mod} k $) in the series of natural numbers. It is believed (the extended Riemann hypothesis) that, for given relatively prime $ l $ and $ k $ and any integer $ x > 1 $,

$$ \pi ( x ; l , k ) = \frac{\int\limits _ { 2 } ^ { x } \frac{d u }{ \mathop{\rm ln} \ u } }{\phi (k) } + O ( x ^ {1 / 2 + \epsilon } ) , $$

where $ \epsilon > 0 $ is arbitrary, while $ O $ is a magnitude depending on $ k $ and $ \epsilon $.

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 $ 2 \pi $- periodic function $ f $ is piecewise monotone on the segment $ [ - \pi , \pi ] $ 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 $ f (x) $ at each continuity point and to $ [ f ( x + 0 ) + f ( x - 0 ) ]/ 2 $ 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)
How to Cite This Entry:
Dirichlet theorem. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Dirichlet_theorem&oldid=33718
This article was adapted from an original article by T.P. Lukashenko (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article