A Dirichlet L-series is a series of the form

where the number theoretic character is an integer function with period k, are called Dirichlet L-series. These series are very important in additive number theory (they were used, for instance, to prove Dirichlet's theorem), and have a close connection with modular forms. Dirichlet L-series can be written as sums of Lerch transcendents with z a power of .

The generalized Riemann hypothesis conjectures that neither the Riemann zeta function nor any Dirichlet L-series has a zero with real part larger than 1/2.

The Dirichlet lambda function

Dirichlet beta function

and Riemann zeta function

are all Dirichlet L-series (Borwein and Borwein 1987, p. 289).

Hecke found a remarkable connection between each modular form with Fourier series

and the Dirichlet L-series

This Dirichlet series converges absolutely for (if f is a cusp form) and if f is not a cusp form. In particular, if the coefficients c(n) satisfy the multiplicative property

then the Dirichlet L-series will have a representation of the form

which is absolutely convergent with the Dirichlet series (Apostol 1997, pp. 136-137). In addition, let be an even integer, then can be analytically continued beyond the line such that

1. If , then is an entire function of s,

2. If , is analytic for all s except a single simple pole at s = k with complex residue

(9)

where is the gamma function, and

3. satisfies

(10)

(Apostol 1997, p. 137).

The number theoretic character is called primitive if the conductor . Otherwise, is imprimitive. A primitive L-series modulo k is then defined as one for which is primitive. All imprimitive L-series can be expressed in terms of primitive L-series.

Let P = 1 or , where are distinct odd primes. Then there are three possible types of primitive L-series with real coefficients. The requirement of real coefficients restricts the number theoretic character to for all k and n. The three type are then

1. If k = P (e.g., k = 1, 3, 5, ...) or (e.g., k = 4, 12, 20, ...), there is exactly one primitive L-series.

2. If (e.g., k = 8, 24, ...), there are two primitive L-series.

3. If , or where (e.g., k = 2, 6, 9, ...), there are no primitive L-series

(Zucker and Robertson 1976). All primitive L-series are algebraically independent and divide into two types according to

Primitive L-series of these types are denoted . For a primitive L-series with real number theoretic character, if k = P, then

If , then

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

The first few primitive negative L-series are , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , ... (Sloane's A003657), corresponding to the negated discriminants of imaginary quadratic fields. The first few primitive positive L-series are , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , ... (Sloane's A003658).

The Kronecker symbol is a real number theoretic character modulo d, and is in fact essentially the only type of real primitive number theoretic character mod d (Ayoub 1963). Therefore,

where is the Kronecker symbol (Borwein and Borwein 1986, p. 293).

Since the Kronecker symbols are periodic with period d, these equations can be written in the form of sums, each of which can be expressed in terms of the polygamma function , giving

Some specific values of primitive L-series are

The functional equations for are

For m a positive integer

where R and are rational numbers. Nothing general appears to be known about or , although it is possible to express all in terms of known transcendentals (Zucker and Robertson 1976).

can be expressed in terms of transcendentals by

where h(d) is the class number and is the Dirichlet structure constant.

No general forms are known for and in terms of known transcendentals. Special examples include

where K is defined as Catalan's constant, and

(Sloane's A103133), where is the trigamma function and is the dilogarithm, where identity (41) was conjectured by Bailey and Borwein (2004).

 

Dirichlet Beta Function, Dirichlet Eta Function, Dirichlet Series, Double Sum, Generalized Riemann Hypothesis, Hecke L-Series, Modular Form, Petersson Conjecture