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''(<img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d032/d032810/d0328101.png" />)''
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A function <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d032/d032810/d0328102.png" /> on the set of integers that satisfies the following conditions:
+
{{TEX|auto}}
 +
{{TEX|done}}
  
<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/d032810/d0328103.png" /></td> </tr></table>
+
''( $  \mathop{\rm mod}  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/d032810/d0328104.png" /></td> </tr></table>
+
A function  $  \chi ( n) = \chi ( n , k) $
 +
on the set of integers that satisfies the following conditions:
  
<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/d032810/d0328105.png" /></td> </tr></table>
+
$$
 +
\chi ( n)  \not\equiv  0 ,
 +
$$
  
In other words, a Dirichlet character <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d032/d032810/d0328106.png" /> is an arithmetic function that is not identically equal to zero, and that is totally multiplicative and periodic with the period <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d032/d032810/d0328107.png" />.
+
$$
 +
\chi ( n) \chi ( l)  =  \chi ( n l) ,
 +
$$
 +
 
 +
$$
 +
\chi ( n)  =  \chi ( n + k ) .
 +
$$
 +
 
 +
In other words, a Dirichlet character $  \mathop{\rm mod}  k $
 +
is an arithmetic function that is not identically equal to zero, and that is totally multiplicative and periodic with the period $  k $.
  
 
The concept of a Dirichlet character was introduced by P.G.L. Dirichlet in the context of his study of the law of the distribution of primes in arithmetic progressions. He developed the fundamental principles of the theory of Dirichlet characters [[#References|[2]]]–[[#References|[8]]], starting from their direct construction.
 
The concept of a Dirichlet character was introduced by P.G.L. Dirichlet in the context of his study of the law of the distribution of primes in arithmetic progressions. He developed the fundamental principles of the theory of Dirichlet characters [[#References|[2]]]–[[#References|[8]]], starting from their direct construction.
Line 15: Line 35:
 
Let
 
Let
  
<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/d032810/d0328108.png" /></td> </tr></table>
+
$$
 +
= 2  ^  \alpha  p _ {1} ^ {\alpha _ {1} } \dots p _ {r} ^ {\alpha _ {r} }
 +
$$
 +
 
 +
be the canonical factorization of  $  k $,
 +
let  $  n $
 +
be an integer which is relatively prime to  $  k $,
 +
$  ( n , k) = 1 $;  
 +
set  $  C = C _ {0} = 1 $
 +
if  $  \alpha = 0 $
 +
or  $  \alpha = 1 $
 +
and  $  C = 2, C _ {0} = 2 ^ {\alpha - 2 } $
 +
if  $  \alpha \geq  2 $;  
 +
let  $  C _ {1} = \phi ( p _ {1} ^ {\alpha _ {1} } ) \dots C _ {r} = \phi ( p _ {r} ^ {\alpha _ {r} } ) $,
 +
where  $  \phi $
 +
is Euler's function. Further, let  $  \gamma , \gamma _ {0} \dots \gamma _ {r} $
 +
be the system of indices of  $  n  \mathop{\rm mod}  k $,
 +
i.e. the system of least non-negative integers satisfying the congruences
 +
 
 +
$$
 +
n  \equiv  ( - 1 )  ^  \gamma  5 ^ {\gamma _ {0} }  (  \mathop{\rm mod}  2  ^  \alpha  ) ,\  n  \equiv  g _ {j} ^ {\gamma _ {j} }  (  \mathop{\rm mod}  p _ {j} ^ {\alpha _ {j} } ) ,\  j = 1 \dots r ,
 +
$$
 +
 
 +
where  $  g _ {j} $
 +
is the smallest primitive root  $  \mathop{\rm mod}  p _ {j} ^ {\alpha _ {j} } $.
 +
Let  $  \epsilon , \epsilon _ {0} \dots \epsilon _ {r} $
 +
be roots of unity of respective orders  $  C , C _ {0} \dots C _ {r} $.  
 +
The function
 +
 
 +
$$
 +
\chi ( n)  = \
 +
\left \{
 +
 
 +
\begin{array}{ll}
 +
\epsilon  ^  \gamma  \epsilon _ {0} ^ {\gamma _ {0} } \dots \epsilon _ {r} ^ {\gamma _ {r} }  &\
 +
\textrm{ if }  ( n , k ) = 1 ,  \\
 +
0  & \textrm{ if }  ( n , k ) \neq 1 ,  \\
 +
\end{array}
 +
 
 +
\right .$$
 +
 
 +
defined on the set of all natural numbers, is a Dirichlet character  $  (  \mathop{\rm mod}  k ) $.  
 +
Inspection of all possible choices of  $  \epsilon , \epsilon _ {0} \dots \epsilon _ {r} $
 +
yields
 +
 
 +
$$
 +
\phi ( 2  ^  \alpha  ) \phi ( p _ {1} ^ {\alpha _ {1} } ) \dots
 +
\phi ( p _ {r} ^ {\alpha _ {r} } )  =  \phi ( k)
 +
$$
  
be the canonical factorization of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d032/d032810/d0328109.png" />, let <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d032/d032810/d03281010.png" /> be an integer which is relatively prime to <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d032/d032810/d03281011.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d032/d032810/d03281012.png" />; set <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d032/d032810/d03281013.png" /> if <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d032/d032810/d03281014.png" /> or <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d032/d032810/d03281015.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d032/d032810/d03281016.png" /> if <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d032/d032810/d03281017.png" />; let <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d032/d032810/d03281018.png" />, where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d032/d032810/d03281019.png" /> is Euler's function. Further, let <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d032/d032810/d03281020.png" /> be the system of indices of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d032/d032810/d03281021.png" />, i.e. the system of least non-negative integers satisfying the congruences
+
different functions  $  \chi $,  
 +
i.e. Dirichlet characters  $  \mathop{\rm mod}  k $.  
 +
The character with  $  \epsilon = \epsilon _ {0} = \dots = \epsilon _ {r} = 1 $
 +
is known as the principal character and is denoted by  $  \chi _ {0} $:
  
<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/d032810/d03281022.png" /></td> </tr></table>
+
$$
 +
\chi _ {0} ( n)  = \left \{
 +
\begin{array}{ll}
 +
1  &\textrm{ if }  ( n , k ) = 1 ,  \\
 +
0   &\textrm{ if }  ( n , k ) \neq 1 . \\
 +
\end{array}
  
where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d032/d032810/d03281023.png" /> is the smallest primitive root <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d032/d032810/d03281024.png" />. Let <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d032/d032810/d03281025.png" /> be roots of unity of respective orders <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d032/d032810/d03281026.png" />. The function
+
\right .$$
  
<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/d032810/d03281027.png" /></td> </tr></table>
+
For any natural numbers  $  n $,
 +
$  l $
 +
and  $  k $,
 +
one has
  
defined on the set of all natural numbers, is a Dirichlet character <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d032/d032810/d03281028.png" />. Inspection of all possible choices of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d032/d032810/d03281029.png" /> yields
+
$$
 +
\chi ( n) \chi ( l)  = \chi ( n l ) ;
 +
$$
  
<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/d032810/d03281030.png" /></td> </tr></table>
+
$$
 +
\chi ( n)  = \chi ( l) \  \textrm{ if }  n \equiv l  (  \mathop{\rm mod}  k ) ;
 +
$$
  
different functions <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d032/d032810/d03281031.png" />, i.e. Dirichlet characters <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d032/d032810/d03281032.png" />. The character with <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d032/d032810/d03281033.png" /> is known as the principal character and is denoted by <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d032/d032810/d03281034.png" />:
+
$$
 +
\chi ( 1)  = 1 ;
 +
$$
  
<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/d032810/d03281035.png" /></td> </tr></table>
+
$$
 +
\chi ( n , k )  = \chi ( n , 2  ^  \alpha  ) \chi ( n , p _ {1} ^ {\alpha _ {1} } ) \dots \chi ( n , p _ {r} ^ {\alpha _ {r} } ) .
 +
$$
  
For any natural numbers <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d032/d032810/d03281036.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d032/d032810/d03281037.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d032/d032810/d03281038.png" />, one has
+
If  $  \chi ( n) $
 +
is a Dirichlet character  $  (  \mathop{\rm mod}  k ) $,  
 +
the complex conjugate function  $  \overline \chi \; ( n) $
 +
is also a Dirichlet character  $  (  \mathop{\rm mod}  k ) $;
 +
and
  
<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/d032810/d03281039.png" /></td> </tr></table>
+
$$
 +
\chi ^ {\phi ( k) } ( n)  = \chi _ {0} ( n) .
 +
$$
  
<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/d032810/d03281040.png" /></td> </tr></table>
+
The smallest positive number  $  \nu $
 +
that satisfies the equation  $  \chi  ^  \nu  ( n) = \chi _ {0} ( n) $
 +
is called the order of the Dirichlet character. For  $  \nu = 1 $
 +
there exists only the character  $  \chi _ {0} $.
 +
If  $  \nu = 2 $,
 +
$  \chi ( n) $
 +
may assume the values 0 and  $  \pm  1 $
 +
only; such Dirichlet characters are known as real or quadratic. If  $  \nu \geq  3 $,
 +
the Dirichlet character is said to be complex.  $  \chi ( n) $
 +
is called even or odd, depending on whether  $  \chi ( - 1 ) = 1 $
 +
or  $  \chi ( - 1 ) = - 1 $.  
 +
The principal properties of Dirichlet characters are expressed by the formulas
  
<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/d032810/d03281041.png" /></td> </tr></table>
+
$$
 +
\sum _ {n  \mathop{\rm mod}  k } \chi ( n)  = \left \{
 +
\begin{array}{ll}
 +
\phi ( k)  & \textrm{ if }  \chi = \chi _ {0} ,  \\
 +
0  & \textrm{ if }  \chi \not\equiv
 +
\chi _ {0} ; \\
 +
\end{array}
  
<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/d032810/d03281042.png" /></td> </tr></table>
+
\right .$$
  
If <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d032/d032810/d03281043.png" /> is a Dirichlet character <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d032/d032810/d03281044.png" />, the complex conjugate function <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d032/d032810/d03281045.png" /> is also a Dirichlet character <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d032/d032810/d03281046.png" />; and
+
$$
 +
\sum _ {\chi  \mathop{\rm mod}  k } \chi ( l)
 +
= \left \{
 +
\begin{array}{ll}
 +
\phi ( k)  &\
 +
\textrm{ if }  l \equiv 1  (  \mathop{\rm mod}  k ) ,  \\
 +
0 &\
 +
\textrm{ if }  l \not\equiv 1  (  \mathop{\rm mod}  k ) , \\
 +
\end{array}
  
<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/d032810/d03281047.png" /></td> </tr></table>
+
\right .$$
  
The smallest positive number <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d032/d032810/d03281048.png" /> that satisfies the equation <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d032/d032810/d03281049.png" /> is called the order of the Dirichlet character. For <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d032/d032810/d03281050.png" /> there exists only the character <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d032/d032810/d03281051.png" />. If <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d032/d032810/d03281052.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d032/d032810/d03281053.png" /> may assume the values 0 and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d032/d032810/d03281054.png" /> only; such Dirichlet characters are known as real or quadratic. If <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d032/d032810/d03281055.png" />, the Dirichlet character is said to be complex. <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d032/d032810/d03281056.png" /> is called even or odd, depending on whether <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d032/d032810/d03281057.png" /> or <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d032/d032810/d03281058.png" />. The principal properties of Dirichlet characters are expressed by the formulas
+
where in the first formula  $  n $
 +
ranges over a complete residue system  $  (  \mathop{\rm mod}  k ) $,  
 +
and  $  \chi $
 +
in the second formula ranges over all  $  \phi ( k) $
 +
characters $  (  \mathop{\rm mod}  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/d032810/d03281059.png" /></td> </tr></table>
+
If  $  ( l , k ) = 1 $,
 +
the formula
  
<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/d032810/d03281060.png" /></td> </tr></table>
+
$$
  
where in the first formula <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d032/d032810/d03281061.png" /> ranges over a complete residue system <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d032/d032810/d03281062.png" />, and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d032/d032810/d03281063.png" /> in the second formula ranges over all <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d032/d032810/d03281064.png" /> characters <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d032/d032810/d03281065.png" />.
+
\frac{1}{\phi ( k) }
 +
\sum _ {\chi  \mathop{\rm mod}  k }
 +
\chi ( n) \overline \chi \; ( l)  = \left \{
 +
\begin{array}{ll}
 +
1  & \textrm{ if }  n \equiv l  (  \mathop{\rm mod}  k ) , \\
 +
0 & \textrm{ if }  n \not\equiv l  (  \mathop{\rm mod}  k)  \\
 +
\end{array}
  
If <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d032/d032810/d03281066.png" />, the formula
+
\right .$$
  
<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/d032810/d03281067.png" /></td> </tr></table>
+
holds. It is called the orthogonality property of Dirichlet characters. It is one of the fundamental formulas for Dirichlet characters and is used in investigating various types of arithmetic progressions  $  \{ k \nu + l,  \nu = 0 , 1 ,\dots \} $.
 +
In the theory and applications of Dirichlet characters other important concepts are the conductor of a character and primitive characters. Let  $  \chi ( n , k ) $
 +
be an arbitrary non-principal character  $  (  \mathop{\rm mod}  k ) $.  
 +
If, for the values  $  n $
 +
satisfying  $  ( n, k) = 1 $,
 +
the number  $  k $
 +
is the smallest period of  $  \chi ( n , k) $,
 +
$  k $
 +
is said to be the conductor of the character  $  \chi $,
 +
while the character  $  \chi $
 +
itself is known as a primitive character  $  (  \mathop{\rm mod}  k ) $.  
 +
Otherwise there exists a unique number  $  k _ {1} > 1 $
 +
dividing  $  k $,
 +
$  k _ {1} < k $,
 +
and a primitive character  $  \chi _ {1} $(
 +
$  \mathop{\rm mod}  k _ {1} $)
 +
such that
  
holds. It is called the orthogonality property of Dirichlet characters. It is one of the fundamental formulas for Dirichlet characters and is used in investigating various types of arithmetic progressions <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d032/d032810/d03281068.png" />. In the theory and applications of Dirichlet characters other important concepts are the conductor of a character and primitive characters. Let <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d032/d032810/d03281069.png" /> be an arbitrary non-principal character <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d032/d032810/d03281070.png" />. If, for the values <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d032/d032810/d03281071.png" /> satisfying <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d032/d032810/d03281072.png" />, the number <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d032/d032810/d03281073.png" /> is the smallest period of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d032/d032810/d03281074.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d032/d032810/d03281075.png" /> is said to be the conductor of the character <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d032/d032810/d03281076.png" />, while the character <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d032/d032810/d03281077.png" /> itself is known as a primitive character <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d032/d032810/d03281078.png" />. Otherwise there exists a unique number <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d032/d032810/d03281079.png" /> dividing <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d032/d032810/d03281080.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d032/d032810/d03281081.png" />, and a primitive character <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d032/d032810/d03281082.png" /> (<img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d032/d032810/d03281083.png" />) such that
+
$$
 +
\chi ( n , k)  = \left \{
 +
\begin{array}{ll}
 +
\chi _ {1} ( n , k )  & \textrm{ if }  ( n , k) = 1 , \\
 +
0 & \textrm{ if }  ( n , k) \neq 1 . \\
 +
\end{array}
  
<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/d032810/d03281084.png" /></td> </tr></table>
+
\right .$$
  
In such a case <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d032/d032810/d03281085.png" /> is said to be the imprimitive character of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d032/d032810/d03281086.png" /> (<img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d032/d032810/d03281087.png" />), and one says that <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d032/d032810/d03281088.png" /> induces <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d032/d032810/d03281089.png" />. In this way many problems on characters are reduced to problems on primitive characters.
+
In such a case $  \chi ( n , k) $
 +
is said to be the imprimitive character of $  \chi _ {1} $(
 +
$  \mathop{\rm mod}  k _ {1} $),  
 +
and one says that $  \chi _ {1} $
 +
induces $  \chi $.  
 +
In this way many problems on characters are reduced to problems on primitive characters.
  
A character <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d032/d032810/d03281090.png" /> is primitive if and only if for any <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d032/d032810/d03281091.png" /> that divides <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d032/d032810/d03281092.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d032/d032810/d03281093.png" />, there exists an <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d032/d032810/d03281094.png" /> that satisfies the conditions
+
A character $  \chi ( n , k) $
 +
is primitive if and only if for any d $
 +
that divides $  k $,
 +
d < k $,  
 +
there exists an $  a $
 +
that satisfies the conditions
  
<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/d032810/d03281095.png" /></td> </tr></table>
+
$$
 +
a  \equiv  1  (  \mathop{\rm mod}  d ) ,\  \chi ( a , k)  \neq  0 , 1 .
 +
$$
  
The analytic theory extensively employs Gauss sums, which are defined for <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d032/d032810/d03281096.png" /> (<img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d032/d032810/d03281097.png" />) by the equality:
+
The analytic theory extensively employs Gauss sums, which are defined for $  \chi $(
 +
$  \mathop{\rm mod}  k $)  
 +
by the equality:
  
<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/d032810/d03281098.png" /></td> </tr></table>
+
$$
 +
\tau ( \chi )  = \sum _ {m= 1 } ^ { k }  \chi ( m) e ^ {2 \pi i m / k } .
 +
$$
  
For a primitive character <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d032/d032810/d03281099.png" /> (<img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d032/d032810/d032810100.png" />) one has
+
For a primitive character $  \chi $(
 +
$  \mathop{\rm mod}  k $)  
 +
one has
  
<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/d032810/d032810101.png" /></td> </tr></table>
+
$$
 +
| \tau ( \chi ) |  = k  ^ {1/2} .
 +
$$
  
Moreover, the following expansion of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d032/d032810/d032810102.png" /> is valid:
+
Moreover, the following expansion of $  \chi ( n) $
 +
is valid:
  
<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/d032810/d032810103.png" /></td> </tr></table>
+
$$
 +
\chi ( n)  =
 +
\frac{1}{\tau ( \overline \chi \; ) }
 +
\sum _ {m = 1 } ^ { k }  \overline \chi \; ( m) e ^ {2 \pi i n m / k } .
 +
$$
  
 
One of the principal problems in the theory of Dirichlet characters is the problem of estimating character sums
 
One of the principal problems in the theory of Dirichlet characters is the problem of estimating character sums
  
<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/d032810/d032810104.png" /></td> </tr></table>
+
$$
 +
S ( N; M)  = \sum _ {M < n \leq  N } \chi ( n) ,
 +
$$
  
where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d032/d032810/d032810105.png" /> is a non-principal character <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d032/d032810/d032810106.png" />. One has Vinogradov's estimate
+
where $  \chi $
 +
is a non-principal character $  \mathop{\rm mod}  k $.  
 +
One has Vinogradov's estimate
  
<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/d032810/d032810107.png" /></td> </tr></table>
+
$$
 +
S ( N; M)  \ll  \sqrt {k }  \mathop{\rm ln}  k .
 +
$$
  
 
It was found [[#References|[7]]] that
 
It was found [[#References|[7]]] that
  
<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/d032810/d032810108.png" /></td> </tr></table>
+
$$
 +
S ( N; M)  \ll  k ^ {( r + 1 ) / 4 r  ^ {2} }
 +
( N - M) ^ {1 - ( 1 / r ) }  \mathop{\rm ln}  k ,\  r = 1 , 2 \dots
 +
$$
  
where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d032/d032810/d032810109.png" /> is a prime. If <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d032/d032810/d032810110.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d032/d032810/d032810111.png" />, there exists [[#References|[8]]] an infinite sequence of numbers <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d032/d032810/d032810112.png" /> which are modules of a primitive real character <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d032/d032810/d032810113.png" /> for which
+
where $  k $
 +
is a prime. If $  M = 1 $,  
 +
$  N = k/2 $,  
 +
there exists [[#References|[8]]] an infinite sequence of numbers $  k $
 +
which are modules of a primitive real character $  \chi $
 +
for which
  
<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/d032810/d032810114.png" /></td> </tr></table>
+
$$
 +
| S ( N; M) |  \sim 
 +
\frac{2 e  ^  \gamma  } \pi
 +
\sqrt {k } \
 +
{ \mathop{\rm ln}  \mathop{\rm ln} }  k ,
 +
$$
  
where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d032/d032810/d032810115.png" /> is the [[Euler constant|Euler constant]]. This asymptotic equation shows that it is not possible, in general, to strengthen the previous estimates essentially. However, there exists Vinogradov's hypothesis according to which for any <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d032/d032810/d032810116.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d032/d032810/d032810117.png" />,
+
where $  \gamma $
 +
is the [[Euler constant|Euler constant]]. This asymptotic equation shows that it is not possible, in general, to strengthen the previous estimates essentially. However, there exists Vinogradov's hypothesis according to which for any $  \epsilon > 0 $,  
 +
$  1 \leq  M < N $,
  
<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/d032810/d032810118.png" /></td> </tr></table>
+
$$
 +
| S ( N; M) |  \ll  k  ^  \epsilon  ( N - M)  ^ {1/2} .
 +
$$
  
 
A proof of this hypothesis would permit one to solve several major problems in number theory.
 
A proof of this hypothesis would permit one to solve several major problems in number theory.
  
The theory of Dirichlet characters forms the basis of the theory of Dirichlet <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d032/d032810/d032810119.png" />-functions (cf. [[Dirichlet L-function|Dirichlet <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d032/d032810/d032810120.png" />-function]]), and is a special case of the general theory of characters of Abelian groups (cf. [[Character of a group|Character of a group]]).
+
The theory of Dirichlet characters forms the basis of the theory of Dirichlet $  L $-
 +
functions (cf. [[Dirichlet L-function|Dirichlet $  L $-
 +
function]]), and is a special case of the general theory of characters of Abelian groups (cf. [[Character of a group|Character of a group]]).
  
 
====References====
 
====References====
 
<table><TR><TD valign="top">[1]</TD> <TD valign="top">  P.G.L. Dirichlet,  "Vorlesungen über Zahlentheorie" , Vieweg  (1894)</TD></TR><TR><TD valign="top">[2]</TD> <TD valign="top">  I.M. Vinogradov,  "Selected works" , Springer  (1985)  (Translated from Russian)</TD></TR><TR><TD valign="top">[3]</TD> <TD valign="top">  A.A. Karatsuba,  "Fundamentals of analytic number theory" , Moscow  (1975)  (In Russian)</TD></TR><TR><TD valign="top">[4]</TD> <TD valign="top">  K. Prachar,  "Primzahlverteilung" , Springer  (1957)</TD></TR><TR><TD valign="top">[5]</TD> <TD valign="top">  N.G. Chudakov,  "Introductions to the theory of Dirichlet <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d032/d032810/d032810121.png" />-functions" , Moscow-Leningrad  (1947)  (In Russian)</TD></TR><TR><TD valign="top">[6]</TD> <TD valign="top">  H. Davenport,  "Multiplicative number theory" , Springer  (1980)</TD></TR><TR><TD valign="top">[7]</TD> <TD valign="top">  D.A. Burgess,  "Dirichlet characters and polynomials"  ''Proc. Steklov Inst. Math.'' , '''132'''  (1975)  pp. 234–236  ''Trudy Mat. Inst. Steklov.'' , '''132'''  (1973)  pp. 203–205</TD></TR><TR><TD valign="top">[8]</TD> <TD valign="top">  A.F. Lavrik,  "A method for estimating double sums with real quadratic character, and applications"  ''Math. USSR-Izv.'' , '''5''' :  6  (1971)  pp. 1195–1214  ''Izv. Akad. Nauk SSSR Ser. Mat.'' , '''35''' :  6  (1971)  pp. 1189–1207</TD></TR></table>
 
<table><TR><TD valign="top">[1]</TD> <TD valign="top">  P.G.L. Dirichlet,  "Vorlesungen über Zahlentheorie" , Vieweg  (1894)</TD></TR><TR><TD valign="top">[2]</TD> <TD valign="top">  I.M. Vinogradov,  "Selected works" , Springer  (1985)  (Translated from Russian)</TD></TR><TR><TD valign="top">[3]</TD> <TD valign="top">  A.A. Karatsuba,  "Fundamentals of analytic number theory" , Moscow  (1975)  (In Russian)</TD></TR><TR><TD valign="top">[4]</TD> <TD valign="top">  K. Prachar,  "Primzahlverteilung" , Springer  (1957)</TD></TR><TR><TD valign="top">[5]</TD> <TD valign="top">  N.G. Chudakov,  "Introductions to the theory of Dirichlet <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d032/d032810/d032810121.png" />-functions" , Moscow-Leningrad  (1947)  (In Russian)</TD></TR><TR><TD valign="top">[6]</TD> <TD valign="top">  H. Davenport,  "Multiplicative number theory" , Springer  (1980)</TD></TR><TR><TD valign="top">[7]</TD> <TD valign="top">  D.A. Burgess,  "Dirichlet characters and polynomials"  ''Proc. Steklov Inst. Math.'' , '''132'''  (1975)  pp. 234–236  ''Trudy Mat. Inst. Steklov.'' , '''132'''  (1973)  pp. 203–205</TD></TR><TR><TD valign="top">[8]</TD> <TD valign="top">  A.F. Lavrik,  "A method for estimating double sums with real quadratic character, and applications"  ''Math. USSR-Izv.'' , '''5''' :  6  (1971)  pp. 1195–1214  ''Izv. Akad. Nauk SSSR Ser. Mat.'' , '''35''' :  6  (1971)  pp. 1189–1207</TD></TR></table>

Revision as of 19:35, 5 June 2020


( $ \mathop{\rm mod} k $)

A function $ \chi ( n) = \chi ( n , k) $ on the set of integers that satisfies the following conditions:

$$ \chi ( n) \not\equiv 0 , $$

$$ \chi ( n) \chi ( l) = \chi ( n l) , $$

$$ \chi ( n) = \chi ( n + k ) . $$

In other words, a Dirichlet character $ \mathop{\rm mod} k $ is an arithmetic function that is not identically equal to zero, and that is totally multiplicative and periodic with the period $ k $.

The concept of a Dirichlet character was introduced by P.G.L. Dirichlet in the context of his study of the law of the distribution of primes in arithmetic progressions. He developed the fundamental principles of the theory of Dirichlet characters [2][8], starting from their direct construction.

Let

$$ k = 2 ^ \alpha p _ {1} ^ {\alpha _ {1} } \dots p _ {r} ^ {\alpha _ {r} } $$

be the canonical factorization of $ k $, let $ n $ be an integer which is relatively prime to $ k $, $ ( n , k) = 1 $; set $ C = C _ {0} = 1 $ if $ \alpha = 0 $ or $ \alpha = 1 $ and $ C = 2, C _ {0} = 2 ^ {\alpha - 2 } $ if $ \alpha \geq 2 $; let $ C _ {1} = \phi ( p _ {1} ^ {\alpha _ {1} } ) \dots C _ {r} = \phi ( p _ {r} ^ {\alpha _ {r} } ) $, where $ \phi $ is Euler's function. Further, let $ \gamma , \gamma _ {0} \dots \gamma _ {r} $ be the system of indices of $ n \mathop{\rm mod} k $, i.e. the system of least non-negative integers satisfying the congruences

$$ n \equiv ( - 1 ) ^ \gamma 5 ^ {\gamma _ {0} } ( \mathop{\rm mod} 2 ^ \alpha ) ,\ n \equiv g _ {j} ^ {\gamma _ {j} } ( \mathop{\rm mod} p _ {j} ^ {\alpha _ {j} } ) ,\ j = 1 \dots r , $$

where $ g _ {j} $ is the smallest primitive root $ \mathop{\rm mod} p _ {j} ^ {\alpha _ {j} } $. Let $ \epsilon , \epsilon _ {0} \dots \epsilon _ {r} $ be roots of unity of respective orders $ C , C _ {0} \dots C _ {r} $. The function

$$ \chi ( n) = \ \left \{ \begin{array}{ll} \epsilon ^ \gamma \epsilon _ {0} ^ {\gamma _ {0} } \dots \epsilon _ {r} ^ {\gamma _ {r} } &\ \textrm{ if } ( n , k ) = 1 , \\ 0 & \textrm{ if } ( n , k ) \neq 1 , \\ \end{array} \right .$$

defined on the set of all natural numbers, is a Dirichlet character $ ( \mathop{\rm mod} k ) $. Inspection of all possible choices of $ \epsilon , \epsilon _ {0} \dots \epsilon _ {r} $ yields

$$ \phi ( 2 ^ \alpha ) \phi ( p _ {1} ^ {\alpha _ {1} } ) \dots \phi ( p _ {r} ^ {\alpha _ {r} } ) = \phi ( k) $$

different functions $ \chi $, i.e. Dirichlet characters $ \mathop{\rm mod} k $. The character with $ \epsilon = \epsilon _ {0} = \dots = \epsilon _ {r} = 1 $ is known as the principal character and is denoted by $ \chi _ {0} $:

$$ \chi _ {0} ( n) = \left \{ \begin{array}{ll} 1 &\textrm{ if } ( n , k ) = 1 , \\ 0 &\textrm{ if } ( n , k ) \neq 1 . \\ \end{array} \right .$$

For any natural numbers $ n $, $ l $ and $ k $, one has

$$ \chi ( n) \chi ( l) = \chi ( n l ) ; $$

$$ \chi ( n) = \chi ( l) \ \textrm{ if } n \equiv l ( \mathop{\rm mod} k ) ; $$

$$ \chi ( 1) = 1 ; $$

$$ \chi ( n , k ) = \chi ( n , 2 ^ \alpha ) \chi ( n , p _ {1} ^ {\alpha _ {1} } ) \dots \chi ( n , p _ {r} ^ {\alpha _ {r} } ) . $$

If $ \chi ( n) $ is a Dirichlet character $ ( \mathop{\rm mod} k ) $, the complex conjugate function $ \overline \chi \; ( n) $ is also a Dirichlet character $ ( \mathop{\rm mod} k ) $; and

$$ \chi ^ {\phi ( k) } ( n) = \chi _ {0} ( n) . $$

The smallest positive number $ \nu $ that satisfies the equation $ \chi ^ \nu ( n) = \chi _ {0} ( n) $ is called the order of the Dirichlet character. For $ \nu = 1 $ there exists only the character $ \chi _ {0} $. If $ \nu = 2 $, $ \chi ( n) $ may assume the values 0 and $ \pm 1 $ only; such Dirichlet characters are known as real or quadratic. If $ \nu \geq 3 $, the Dirichlet character is said to be complex. $ \chi ( n) $ is called even or odd, depending on whether $ \chi ( - 1 ) = 1 $ or $ \chi ( - 1 ) = - 1 $. The principal properties of Dirichlet characters are expressed by the formulas

$$ \sum _ {n \mathop{\rm mod} k } \chi ( n) = \left \{ \begin{array}{ll} \phi ( k) & \textrm{ if } \chi = \chi _ {0} , \\ 0 & \textrm{ if } \chi \not\equiv \chi _ {0} ; \\ \end{array} \right .$$

$$ \sum _ {\chi \mathop{\rm mod} k } \chi ( l) = \left \{ \begin{array}{ll} \phi ( k) &\ \textrm{ if } l \equiv 1 ( \mathop{\rm mod} k ) , \\ 0 &\ \textrm{ if } l \not\equiv 1 ( \mathop{\rm mod} k ) , \\ \end{array} \right .$$

where in the first formula $ n $ ranges over a complete residue system $ ( \mathop{\rm mod} k ) $, and $ \chi $ in the second formula ranges over all $ \phi ( k) $ characters $ ( \mathop{\rm mod} k ) $.

If $ ( l , k ) = 1 $, the formula

$$ \frac{1}{\phi ( k) } \sum _ {\chi \mathop{\rm mod} k } \chi ( n) \overline \chi \; ( l) = \left \{ \begin{array}{ll} 1 & \textrm{ if } n \equiv l ( \mathop{\rm mod} k ) , \\ 0 & \textrm{ if } n \not\equiv l ( \mathop{\rm mod} k) \\ \end{array} \right .$$

holds. It is called the orthogonality property of Dirichlet characters. It is one of the fundamental formulas for Dirichlet characters and is used in investigating various types of arithmetic progressions $ \{ k \nu + l, \nu = 0 , 1 ,\dots \} $. In the theory and applications of Dirichlet characters other important concepts are the conductor of a character and primitive characters. Let $ \chi ( n , k ) $ be an arbitrary non-principal character $ ( \mathop{\rm mod} k ) $. If, for the values $ n $ satisfying $ ( n, k) = 1 $, the number $ k $ is the smallest period of $ \chi ( n , k) $, $ k $ is said to be the conductor of the character $ \chi $, while the character $ \chi $ itself is known as a primitive character $ ( \mathop{\rm mod} k ) $. Otherwise there exists a unique number $ k _ {1} > 1 $ dividing $ k $, $ k _ {1} < k $, and a primitive character $ \chi _ {1} $( $ \mathop{\rm mod} k _ {1} $) such that

$$ \chi ( n , k) = \left \{ \begin{array}{ll} \chi _ {1} ( n , k ) & \textrm{ if } ( n , k) = 1 , \\ 0 & \textrm{ if } ( n , k) \neq 1 . \\ \end{array} \right .$$

In such a case $ \chi ( n , k) $ is said to be the imprimitive character of $ \chi _ {1} $( $ \mathop{\rm mod} k _ {1} $), and one says that $ \chi _ {1} $ induces $ \chi $. In this way many problems on characters are reduced to problems on primitive characters.

A character $ \chi ( n , k) $ is primitive if and only if for any $ d $ that divides $ k $, $ d < k $, there exists an $ a $ that satisfies the conditions

$$ a \equiv 1 ( \mathop{\rm mod} d ) ,\ \chi ( a , k) \neq 0 , 1 . $$

The analytic theory extensively employs Gauss sums, which are defined for $ \chi $( $ \mathop{\rm mod} k $) by the equality:

$$ \tau ( \chi ) = \sum _ {m= 1 } ^ { k } \chi ( m) e ^ {2 \pi i m / k } . $$

For a primitive character $ \chi $( $ \mathop{\rm mod} k $) one has

$$ | \tau ( \chi ) | = k ^ {1/2} . $$

Moreover, the following expansion of $ \chi ( n) $ is valid:

$$ \chi ( n) = \frac{1}{\tau ( \overline \chi \; ) } \sum _ {m = 1 } ^ { k } \overline \chi \; ( m) e ^ {2 \pi i n m / k } . $$

One of the principal problems in the theory of Dirichlet characters is the problem of estimating character sums

$$ S ( N; M) = \sum _ {M < n \leq N } \chi ( n) , $$

where $ \chi $ is a non-principal character $ \mathop{\rm mod} k $. One has Vinogradov's estimate

$$ S ( N; M) \ll \sqrt {k } \mathop{\rm ln} k . $$

It was found [7] that

$$ S ( N; M) \ll k ^ {( r + 1 ) / 4 r ^ {2} } ( N - M) ^ {1 - ( 1 / r ) } \mathop{\rm ln} k ,\ r = 1 , 2 \dots $$

where $ k $ is a prime. If $ M = 1 $, $ N = k/2 $, there exists [8] an infinite sequence of numbers $ k $ which are modules of a primitive real character $ \chi $ for which

$$ | S ( N; M) | \sim \frac{2 e ^ \gamma } \pi \sqrt {k } \ { \mathop{\rm ln} \mathop{\rm ln} } k , $$

where $ \gamma $ is the Euler constant. This asymptotic equation shows that it is not possible, in general, to strengthen the previous estimates essentially. However, there exists Vinogradov's hypothesis according to which for any $ \epsilon > 0 $, $ 1 \leq M < N $,

$$ | S ( N; M) | \ll k ^ \epsilon ( N - M) ^ {1/2} . $$

A proof of this hypothesis would permit one to solve several major problems in number theory.

The theory of Dirichlet characters forms the basis of the theory of Dirichlet $ L $- functions (cf. Dirichlet $ L $- function), and is a special case of the general theory of characters of Abelian groups (cf. Character of a group).

References

[1] P.G.L. Dirichlet, "Vorlesungen über Zahlentheorie" , Vieweg (1894)
[2] I.M. Vinogradov, "Selected works" , Springer (1985) (Translated from Russian)
[3] A.A. Karatsuba, "Fundamentals of analytic number theory" , Moscow (1975) (In Russian)
[4] K. Prachar, "Primzahlverteilung" , Springer (1957)
[5] N.G. Chudakov, "Introductions to the theory of Dirichlet -functions" , Moscow-Leningrad (1947) (In Russian)
[6] H. Davenport, "Multiplicative number theory" , Springer (1980)
[7] D.A. Burgess, "Dirichlet characters and polynomials" Proc. Steklov Inst. Math. , 132 (1975) pp. 234–236 Trudy Mat. Inst. Steklov. , 132 (1973) pp. 203–205
[8] A.F. Lavrik, "A method for estimating double sums with real quadratic character, and applications" Math. USSR-Izv. , 5 : 6 (1971) pp. 1195–1214 Izv. Akad. Nauk SSSR Ser. Mat. , 35 : 6 (1971) pp. 1189–1207
How to Cite This Entry:
Dirichlet character. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Dirichlet_character&oldid=46716
This article was adapted from an original article by A.F. Lavrik (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article