## Dirichlet L-Series

Series of the form

 (1)

where the Character (Number Theory) is an Integer function with period . These series appear in number theory (they were used, for instance, to prove Dirichlet's Theorem) and can be written as sums of Lerch Transcendents with a Power of . The Dirichlet Eta Function
 (2)

(for ) and Dirichlet Beta Function
 (3)

and Riemann Zeta Function
 (4)

are Dirichlet series (Borwein and Borwein 1987, p. 289). is called primitive if the Conductor . Otherwise, is imprimitive. A primitive -series modulo is then defined as one for which is primitive. All imprimitive -series can be expressed in terms of primitive -series.

Let or , where are distinct Odd Primes. Then there are three possible types of primitive -series with Real Coefficients. The requirement of Real Coefficients restricts the Character to for all and . The three type are then

1. If (e.g., , 3, 5, ...) or (e.g., , 12, 20, ...), there is exactly one primitive -series.

2. If (e.g., , 24, ...), there are two primitive -series.

3. If , or where (e.g., , 6, 9, ...), there are no primitive -series
(Zucker and Robertson 1976). All primitive -series are algebraically independent and divide into two types according to
 (5)

Primitive -series of these types are denoted . For a primitive -series with Real Character (Number Theory), if , then
 (6)

If , then
 (7)

and if , then there is a primitive function of each type (Zucker and Robertson 1976).

The first few primitive Negative -series are , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , ... (Sloane's A003657), corresponding to the negated discriminants of imaginary quadratic fields. The first few primitive Positive -series are , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , ... (Sloane's A046113).

The Kronecker Symbol is a Real Character modulo , and is in fact essentially the only type of Real primitive Character (Ayoub 1963). Therefore,

 (8) (9)

where is the Kronecker Symbol. The functional equations for are
 (10) (11)

For a Positive Integer
 (12) (13) (14) (15) (16) (17)

where and are Rational Numbers. can be expressed in terms of transcendentals by
 (18)

where is the Class Number and is the Dirichlet Structure Constant. Some specific values of primitive -series are

No general forms are known for and in terms of known transcendentals. For example,
 (19)

where is defined as Catalan's Constant.

References

Ayoub, R. G. An Introduction to the Analytic Theory of Numbers. Providence, RI: Amer. Math. Soc., 1963.

Borwein, J. M. and Borwein, P. B. Pi & the AGM: A Study in Analytic Number Theory and Computational Complexity. New York: Wiley, 1987.

Buell, D. A. Small Class Numbers and Extreme Values of -Functions of Quadratic Fields.'' Math. Comput. 139, 786-796, 1977.

Ireland, K. and Rosen, M. Dirichlet -Functions.'' Ch. 16 in A Classical Introduction to Modern Number Theory, 2nd ed. New York: Springer-Verlag, pp. 249-268, 1990.

Sloane, N. J. A. Sequences A046113 and A003657/M2332 in An On-Line Version of the Encyclopedia of Integer Sequences.'' http://www.research.att.com/~njas/sequences/eisonline.html and Sloane, N. J. A. and Plouffe, S. The Encyclopedia of Integer Sequences. San Diego: Academic Press, 1995.

Weisstein, E. W. Class Numbers.'' Mathematica notebook ClassNumbers.m.

Zucker, I. J. and Robertson, M. M. Some Properties of Dirichlet -Series.'' J. Phys. A: Math. Gen. 9, 1207-1214, 1976.