Dirichlet L-Series

A Dirichlet -series is a series of the form

where is a number theoretic character modulo a positive integer . The series converges absolutely for . 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 -series can be written as sums of Lerch transcendents with a power of .

Dirichlet -series is implemented in the Wolfram Language as DirichletL[k, j, s] for the Dirichlet character with modulus and index .

The generalized Riemann hypothesis conjectures that neither the Riemann zeta function nor any Dirichlet -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 -series (Borwein and Borwein 1987, p. 289).

More general Dirichlet series arise from modular forms (Hecke 1936). Let be a modular form of weight for the modular group Gamma, with an integer and Fourier series

and associate to it the Dirichlet series

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

then the Dirichlet series will have a representation of the form

which is absolutely convergent with the Dirichlet series (Apostol 1997, pp. 136-137). In addition, has an analytic continuation satisfying

1. If , then is an entire function of ,

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

where is the gamma function, and

3. satisfies

(Apostol 1997, p. 137).

The conductor of a number theoretic character is the smallest positive divisor of such that is induced by a number theoretic character modulo . It is called primitive if and imprimitive otherwise. A primitive L-series modulo is one for which is primitive. Every imprimitive -series can be expressed in terms of a primitive L-series.

Let 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 values 0, 1, and , with when is not relatively prime to . The three types are then

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

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

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

(Zucker and Robertson 1976). Distinct primitive L-series, viewed as functions of , 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 , then

If , then

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

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

The Kronecker symbol is a real number theoretic character modulo when is a fundamental discriminant, and is primitive. Every real primitive number theoretic character arises this way, including the trivial character for (Ayoub 1963). Therefore,

where is the Kronecker symbol (Borwein and Borwein 1987, p. 293).

For a fundamental discriminant , the Kronecker symbol is periodic with period . Grouping terms by residue class gives, for integers , a finite sum of polygamma functions,

The same identity holds at after cancellation of the divergent residue-class terms.

The functional equations for are

(Borwein and Borwein 1987, p. 303), with removable singularities interpreted by limits.

For a positive conductor of the indicated parity and a positive integer

where and are rational numbers defined by the corresponding positive-argument values above. All nonprincipal can be expressed in terms of known transcendentals (Zucker and Robertson 1976).

For a positive fundamental discriminant , can be expressed in terms of transcendentals by

where is the class number and is the Dirichlet structure constant.

No general forms are known for and in terms of known transcendentals. Edwards (2000) gives several examples of special cases of . A number of primitive series are given by

and for are given by

where is Catalan's constant, is the trigamma function, and is the dilogarithm.

Some cubic values are

These follow from the Fourier series of the Bernoulli polynomials and the functional equation for primitive Dirichlet -series (NIST, §§24.8 and 25.15).

Bailey and Borwein (Bailey and Borwein 2005; Bailey et al. 2006a, pp. 5 and 62; Bailey et al. 2006b; Bailey and Borwein 2008; Coffey 2008) conjectured the relation actually in effect proved by Zagier (1986) nearly twenty years earlier (M. Coffey, pers. comm., Mar. 30, 2009) that is also given by

where the latter expressions are due to Coffey (2008ab), with

Kumar and Tosi (2026, Theorem 1) obtain a quantitative irrationality result for each nontrivial real Dirichlet character . Among the noncritical values with , at least are irrational for all sufficiently large , where depends on the conductor. Noncritical here means that has parity opposite to that of . The result counts irrational values collectively and does not identify a specified individual value as irrational.