Dirichlet L-Series

A Dirichlet -series is a series of the form

(1)

where the number theoretic character  is an integer function with period , are called Dirichlet -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 -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

			(2)
			(3)

Dirichlet beta function

			(4)
			(5)

and Riemann zeta function

			(6)
			(7)

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

Hecke (1936) found a remarkable connection between each modular form with Fourier series

(8)

and the Dirichlet -series

(9)

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

(10)

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

(11)

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 ,

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

(12)

where  is the gamma function, and

3.  satisfies

(13)

(Apostol 1997, p. 137).

The number theoretic character  is called primitive if the j-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 number theoretic 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

(14)

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

(15)

If , then

(16)

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 , and is in fact essentially the only type of real primitive number theoretic character mod  (Ayoub 1963). Therefore,

(17)

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

For primitive values of , the Kronecker symbols are periodic with period , so  can be written in the form of  sums, each of which can be expressed in terms of the polygamma function , giving

(18)

The functional equations for  are

			(19)
			(20)

(Borwein and Borwein 1986, p. 303).

For  a positive integer

			(21)
			(22)
			(23)
			(24)
			(25)
			(26)

where  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

(27)

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

			(28)
			(29)
			(30)
			(31)
			(32)
			(33)
			(34)
			(35)
			(36)
			(37)
			(38)
			(39)
			(40)
			(41)

and for  are given by

			(42)
			(43)
			(44)
			(45)
			(46)
			(47)
			(48)
			(49)
			(50)
			(51)
			(52)

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

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

			(53)
			(54)
			(55)
			(56)
			(57)

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

			(58)
			(59)
			(60)
			(61)
			(62)

REFERENCES:

Apostol, T. M. Introduction to Analytic Number Theory. New York: Springer-Verlag, 1976.

Apostol, T. M. "Modular Forms and Dirichlet Series" and "Equivalence of Ordinary Dirichlet Series." §6.16 and §8.8 in Modular Functions and Dirichlet Series in Number Theory, 2nd ed. New York: Springer-Verlag, pp. 136-137 and 174-176, 1997.

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

Bailey, D. H. and Borwein, J. M. "Experimental Mathematics: Examples, Methods, and Implications." Not. Amer. Math. Soc. 52, 502-514, 2005.

Bailey, D. H. and Borwein, J. M. "Computer-Assisted Discovery and Proof." In Tapas in Experimental Mathematics (Ed. T. Amdeberhan and V. Moll). Providence, RI: Amer. Math. Soc., 2008.

Bailey, D. H.; Borwein, J. M.; Calkin, N. J.; Girgensohn, R.; Luke, D. R.; and Moll, V. H. Experimental Mathematics in Action. Wellesley, MA: A K Peters, p. 222, 2006a. http://crd.lbl.gov/~dhbailey/expmath/maa-course/hyper-ema.pdf.

Bailey, D. H.; Borwein, J. M.; Kapoor, V.; and Weisstein, E. W. "Ten Problems in Experimental Mathematics." Amer. Math. Monthly 113, 481-509, 2006b.

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.

Coffey, M. W. "Evaluation of a ln tan Integral Arising in Quantum Field Theory." J. Math. Phys. 49, 093508-1-15, 2008a.

Coffey, M. W. "Alternative Evaluation of a ln tan Integral Arising in Quantum Field Theory." Nov. 15, 2008b. http://arxiv.org/abs/0810.5077.

Edwards, H. M. Fermat's Last Theorem : A Genetic Introduction to Algebraic Number Theory. New York: Springer-Verlag, 2000.

Hecke, E. "Über die Bestimmung Dirichletscher Reihen durch ihre Funktionalgleichung." Math. Ann. 112, 664-699, 1936.

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.

Koch, H. "L-Series." Ch. 7 in Number Theory: Algebraic Numbers and Functions. Providence, RI: Amer. Math. Soc., pp. 203-258, 2000.

Shanks, D. and Wrench, J. W. Jr. "The Calculation of Certain Dirichlet Series." Math. Comput. 17, 135-154, 1963.

Shanks, D. and Wrench, J. W. Jr. "Corrigendum to 'The Calculation of Certain Dirichlet Series.' " Math. Comput. 17, 488, 1963.

Sloane, N. J. A. Sequences A003657/M2332, A003658/M3776, and A103133 in "The On-Line Encyclopedia of Integer Sequences."

Zagier, D. "Hyperbolic Manifolds and Special Values of Dedekind Zeta-Functions." Invent. Math. 83, 285-301, 1986.

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