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Dirichlet L-Series

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A Dirichlet L-series is a series of the form

L_k(s,chi)=sum_(n=1)^inftychi_k(n)n^(-s),
(1)

where the number theoretic character chi_k(n) 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 e^(2pii/k).

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

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

lambda(s)=sum_(n=0)^(infty)1/((2n+1)^s)
(2)
=(1-2^(-s))zeta(s),
(3)

Dirichlet beta function

L_(-4)(s)=sum_(n=0)^(infty)((-1)^n)/((2n+1)^s),
(4)
=beta(s)
(5)

and Riemann zeta function

L_(+1)(s)=zeta(s)
(6)
=sum_(n=1)^(infty)1/(n^s)
(7)

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

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

f(tau)=c(0)+sum_(n=1)^inftyc(n)e^(2piintau)
(8)

and the Dirichlet L-series

phi(s)=sum_(n=1)^infty(c(n))/(n^s)
(9)

This Dirichlet series converges absolutely for sigma=R[s]>k+1 (if f is a cusp form) and sigma>2k if f is not a cusp form. In particular, if the coefficients c(n) satisfy the multiplicative property

c(m)c(n)=sum_(d|(m,n))d^(2k-1)c((mn)/(d^2)),
(10)

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

phi(s)=product_(p)1/(1-c(p)p^(-s)+p^(2k-1)p^(-2s)),
(11)

which is absolutely convergent with the Dirichlet series (Apostol 1997, pp. 136-137). In addition, let k>=4 be an even integer, then phi(s) can be analytically continued beyond the line sigma=k such that

1. If c(0)=0, then phi(s) is an entire function of s,

2. If c(0)!=0, phi(s) is analytic for all s except a single simple pole at s=k with complex residue

((-1)^(k/2)c(0)(2pi)^k)/(Gamma(k)),
(12)

where Gamma(k) is the gamma function, and

3. phi(s) satisfies

(2pi)^(-s)Gamma(s)phi(s)=(-1)^(k/2)(2pi)^(s-k)Gamma(k-s)phi(k-s)
(13)

(Apostol 1997, p. 137).

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

Let P=1 or P=product_(i=1)^(t)p_i, where p_i 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 chi_k(n)=+/-1 for all k and n. The three type are then

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

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

3. If k=2P,Pp_i, or 2^alphaP where alpha>3 (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

chi_k(k-1)=+/-1.
(14)

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

L_k={L_(-k) if P=3 (mod 4); L_k if P=1 (mod 4).
(15)

If k=4P, then

L_k={L_(-k) if P=1 (mod 4); L_k if P=3 (mod 4),
(16)

and if k=8P, then there is a primitive function of each type (Zucker and Robertson 1976).

The first few primitive negative L-series are L_(-3), L_(-4), L_(-7), L_(-8), L_(-11), L_(-15), L_(-19), L_(-20), L_(-23), L_(-24), L_(-31), L_(-35), L_(-39), L_(-40), L_(-43), L_(-47), L_(-51), L_(-52), L_(-55), L_(-56), L_(-59), L_(-67), L_(-68), L_(-71), L_(-79), L_(-83), L_(-84), L_(-87), L_(-88), L_(-91), L_(-95), ... (OEIS A003657), corresponding to the negated discriminants of imaginary quadratic fields. The first few primitive positive L-series are L_(+1), L_(+5), L_(+8), L_(+12), L_(+13), L_(+17), L_(+21), L_(+24), L_(+28), L_(+29), L_(+33), L_(+37), L_(+40), L_(+41), L_(+44), L_(+53), L_(+56), L_(+57), L_(+60), L_(+61), L_(+65), L_(+69), L_(+73), L_(+76), L_(+77), L_(+85), L_(+88), L_(+89), L_(+92), L_(+93), L_(+97), ... (OEIS A003658).

The Kronecker symbol (d/n) 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,

L_d(s)=sum_(n=1)^infty(d/n)n^(-s)
(17)

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

For primitive values of d, the Kronecker symbols are periodic with period |d|, so L_d(s) can be written in the form of |d|-1 sums, each of which can be expressed in terms of the polygamma function psi_n(z), giving

L_d(s)=1/((-|d|)^s(s-1)!)sum_(n=1)^(|d|-1)(d/n)psi_(s-1)(n/(|d|)).
(18)

The functional equations for L_+/- are

L_(-d)(s)=2^spi^(s-1)d^(-s+1/2)Gamma(1-s)cos(1/2spi)L_(-d)(1-s)
(19)
L_(+d)(s)=2^spi^(s-1)d^(-s+1/2)Gamma(1-s)sin(1/2spi)L_(+d)(1-s)
(20)

(Borwein and Borwein 1986, p. 303).

For m a positive integer

L_(+d)(-2m)=0
(21)
L_(-d)(1-2m)=0
(22)
L_(+d)(2m)=Rk^(-1/2)pi^(2m)
(23)
L_(-d)(2m-1)=R^'k^(-1/2)pi^(2m-1)
(24)
L_(+d)(1-2m)=((-1)^m(2m-1)!R)/((2k)^(2m-1))
(25)
L_(-d)(-2k)=((-1)^mR^'(2m)!)/((2k)^(2m)),
(26)

where R and R^' are rational numbers. Nothing general appears to be known about L_(-d)(2m) or L_(+d)(2m-1), although it is possible to express all L_+/-(1) in terms of known transcendentals (Zucker and Robertson 1976).

L_(+d)(1) can be expressed in terms of transcendentals by

L_d(1)=h(d)kappa(d),
(27)

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

No general forms are known for L_(-d)(2m) and L_(+d)(2m-1) in terms of known transcendentals. Edwards (2000) gives several examples of special cases of L_d(1). A number of primitive series L_d(1) are given by

L_(-20)(1)=pi/(sqrt(5))
(28)
L_(-15)(1)=(2pi)/(sqrt(15))
(29)
L_(-11)(1)=pi/(sqrt(11))
(30)
L_(-8)(1)=pi/(2sqrt(2))
(31)
L_(-7)(1)=pi/(sqrt(7))
(32)
L_(-4)(1)=1/4pi
(33)
L_(-3)(1)=1/9pisqrt(3)
(34)
L_(+5)(1)=2/5sqrt(5)lnphi
(35)
L_(+8)(1)=(ln(1+sqrt(2)))/(sqrt(2))
(36)
L_(+12)(1)=(ln(2+sqrt(3)))/(sqrt(3))
(37)
L_(+13)(1)=2/(sqrt(13))ln((3+sqrt(13))/2)
(38)
L_(+17)(1)=2/(sqrt(17))ln(4+sqrt(17))
(39)
L_(+21)(1)=2/(sqrt(21))ln((5+sqrt(21))/2)
(40)
L_(+24)(1)=(ln(5+2sqrt(6)))/(sqrt(6)),
(41)

and for L_k(2) are given by

L_(-8)(2)=1/(64)[psi_1(1/8)+psi_1(3/8)-psi_1(5/8)-psi_1(7/8)]
(42)
L_(-7)(2)=1/(49)[psi_1(1/7)+psi_1(2/7)-psi_1(3/7)+psi_1(4/7)
(43)
L_(-4)(2)=K
(44)
L_(-3)(2)=1/9[psi_1(1/3)-psi_1(2/3)]
(45)
L_(+1)(2)=1/6pi^2
(46)
L_(+5)(2)=4/(125)pi^2sqrt(5)
(47)
L_(+8)(2)=1/(16)pi^2sqrt(2)
(48)
L_(+12)(2)=1/(18)pi^2sqrt(3)
(49)
L_(+13)(2)=(4pi^2)/(13sqrt(13))
(50)
L_(+17)(2)=(8pi^2)/(17sqrt(17))
(51)
L_(+21)(2)=(8pi^2)/(21sqrt(21)),
(52)

where K is Catalan's constant, psi_1(z) is the trigamma function, and Li_2(z) 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 L_(-7)(2) is also given by

I_7=(24)/(7sqrt(7))int_(pi/3)^(pi/2)ln|(tanx+sqrt(7))/(tanx-sqrt(7))|dx
(53)
=-4/(7sqrt(7)){9ln2cot^(-1)sqrt(7)+(pi-6cot^(-1)sqrt(7))×ln(sqrt(7)-sqrt(3))-piln(sqrt(3)+sqrt(7))+3i[Li_2((sqrt(7)-sqrt(3))/(sqrt(7)-i))-Li_2((sqrt(7)-sqrt(3))/(sqrt(7)+i))-Li_2((sqrt(7)-i)/(sqrt(3)+sqrt(7)))+Li_2((sqrt(7)+i)/(sqrt(3)+sqrt(7)))]}
(54)
=(24)/(7sqrt(7)){Cl_2(theta_+)+1/2[Cl_2(2omega_+)-Cl_2(2omega_++2theta_+)]}
(55)
=4/(7sqrt(7))[3Cl_2(theta_7)-3Cl_2(2theta_7)+Cl_2(3theta_7)]
(56)
=1.1519254705...
(57)

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

omega_+=tan^(-1)(sqrt(7))-(2pi)/3
(58)
omega_-=-omega_+
(59)
=tan^(-1)((2sqrt(3)-sqrt(7))/5)
(60)
theta_+=tan^(-1)(1/3sqrt(7))
(61)
theta_7=2tan^(-1)(sqrt(7)).
(62)

See also

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

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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 L-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 L-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."Tyagi, S. "Double Exponential Method for Riemann Zeta, Lerch and Dirichlet L-Functions." https://arxiv.org/abs/2203.02509. 7 Mar 2022.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 L-Series." J. Phys. A: Math. Gen. 9, 1207-1214, 1976.

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Dirichlet L-Series

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Weisstein, Eric W. "Dirichlet L-Series." From MathWorld--A Wolfram Web Resource. https://mathworld.wolfram.com/DirichletL-Series.html

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