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''( $   \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 [[#References|[2]]]–[[#References|[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 [[#References|[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 [[#References|[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|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|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====
<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>