Dirichlet character

()

A function on the set of integers that satisfies the following conditions:

In other words, a Dirichlet character is an arithmetic function that is not identically equal to zero, and that is totally multiplicative and periodic with the period .

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

be the canonical factorization of , let be an integer which is relatively prime to , ; set if or and if ; let , where is Euler's function. Further, let be the system of indices of , i.e. the system of least non-negative integers satisfying the congruences

where is the smallest primitive root . Let be roots of unity of respective orders . The function

defined on the set of all natural numbers, is a Dirichlet character . Inspection of all possible choices of yields

different functions , i.e. Dirichlet characters . The character with is known as the principal character and is denoted by :

For any natural numbers , and , one has

If is a Dirichlet character , the complex conjugate function is also a Dirichlet character ; and

The smallest positive number that satisfies the equation is called the order of the Dirichlet character. For there exists only the character . If , may assume the values 0 and only; such Dirichlet characters are known as real or quadratic. If , the Dirichlet character is said to be complex. is called even or odd, depending on whether or . The principal properties of Dirichlet characters are expressed by the formulas

where in the first formula ranges over a complete residue system , and in the second formula ranges over all characters .

If , the formula

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 . In the theory and applications of Dirichlet characters other important concepts are the conductor of a character and primitive characters. Let be an arbitrary non-principal character . If, for the values satisfying , the number is the smallest period of , is said to be the conductor of the character , while the character itself is known as a primitive character . Otherwise there exists a unique number dividing , , and a primitive character () such that

In such a case is said to be the imprimitive character of (), and one says that induces . In this way many problems on characters are reduced to problems on primitive characters.

A character is primitive if and only if for any that divides , , there exists an that satisfies the conditions

The analytic theory extensively employs Gauss sums, which are defined for () by the equality:

For a primitive character () one has

Moreover, the following expansion of is valid:

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

where is a non-principal character . One has Vinogradov's estimate

It was found [7] that

where is a prime. If , , there exists [8] an infinite sequence of numbers which are modules of a primitive real character for which

where 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 , ,

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 -functions (cf. Dirichlet -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