Dirichlet L-Series

Series of the form

$$L_k(s,\chi) \equiv \sum_{n=1}^\infty \chi_k(n)n^{-s},$$ (1)

where the Character (Number Theory) $\chi_k(n)$ is an Integer function with period $m$. 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 $z$ a Power of $e^{2\pi i/m}$. The Dirichlet Eta Function

$$\eta (s)\equiv \sum_{n=1}^\infty {(-1)^{n+1}\over n^s}= (1-2^{1-s})\zeta(s)$$ (2)

(for $s\not=1$) and Dirichlet Beta Function

$$L_{-4}(s)=\beta(s)\equiv \sum_{n=0}^\infty {(-1)^n\over (2n+1)^s}$$ (3)

and Riemann Zeta Function

$$L_{+1}(s)=\zeta(s)$$ (4)

are Dirichlet series (Borwein and Borwein 1987, p. 289). $\chi_k$ is called primitive if the 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 $P=\prod_{i=1}^t p_i$, 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 Character to $\chi_k(n)=\pm 1$ 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 $2^\alpha P$ 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

$$\chi_k(k-1)=\pm 1.$$ (5)

Primitive $L$-series of these types are denoted $L_\pm$. For a primitive $L$-series with Real Character (Number Theory), if $k=P$, then

$$L_k=\cases{ L_{-k} & if $P\equiv 3\ \left({{\rm mod\ } {4}}... ...\cr L_k & if $P\equiv 1\ \left({{\rm mod\ } {4}}\right)$.\cr}$$ (6)

If $k=4P$, then

$$L_k=\cases{ L_{-k} & if $P\equiv 1\ \left({{\rm mod\ } {4}}... ...\cr L_k & if $P\equiv 3\ \left({{\rm mod\ } {4}}\right)$,\cr}$$ (7)

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 $L_{-3}$, $L_{-4}$, $L_{-7}$, $L_{-8}$, $L_{-11}$, $L_{-15}$, $L_{-19}$, $L_{-20}$, $L_{-23}$, $L_{-24}$, $L_{-31}$, $L_{-35}$, $L_{-39}$, $L_{-40}$, $L_{-43}$, $L_{-47}$, $L_{-51}$, $L_{-52}$, $L_{-55}$, $L_{-56}$, $L_{-59}$, $L_{-67}$, $L_{-68}$, $L_{-71}$, $L_{-79}$, $L_{-83}$, $L_{-84}$, $L_{-87}$, $L_{-88}$, $L_{-91}$, $L_{-95}$, ... (Sloane's A003657), corresponding to the negated discriminants of imaginary quadratic fields. The first few primitive Positive $L$-series are $L_{+1}$, $L_{+5}$, $L_{+8}$, $L_{+12}$, $L_{+13}$, $L_{+17}$, $L_{+21}$, $L_{+24}$, $L_{+28}$, $L_{+29}$, $L_{+33}$, $L_{+37}$, $L_{+40}$, $L_{+41}$, $L_{+44}$, $L_{+53}$, $L_{+56}$, $L_{+57}$, $L_{+60}$, $L_{+61}$, $L_{+65}$, $L_{+ 69}$, $L_{+73}$, $L_{+76}$, $L_{+77}$, $L_{+85}$, $L_{+88}$, $L_{+89}$, $L_{+92}$, $L_{+93}$, $L_{+97}$, ... (Sloane's A046113).

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

$$L_{+d}(s) = \sum_{n=1}^\infty (d\vert n)n^{-s}$$ (8)

$$L_{-d}(s) = \sum_{n=1}^\infty (-d\vert n)n^{-s},$$ (9)

where $(d\vert n)$ is the Kronecker Symbol. The functional equations for $L_\pm$ are

$$L_{-k}(s) = 2^s\pi^{s-1}k^{-s+1/2}\Gamma(1-s)\cos({\textstyle{1\over 2}}s\pi) L_{-k}(1-s)$$ (10)

$$L_{+k}(s) = 2^s\pi^{s-1}k^{-s+1/2}\Gamma(1-s)\sin({\textstyle{1\over 2}}s\pi) L_{+k}(1-s).$$ (11)

For $m$ a Positive Integer

$$L_{+k}(-2m) = 0$$ (12)

$$L_{-k}(1-2m) = 0$$ (13)

$$L_{+k}(2m) = Rk^{-1/2}\pi^{2m}$$ (14)

$$L_{-k}(2m-1) = R' k^{-1/2} \pi^{2m-1}$$ (15)

$$L_{+k}(1-2m) = {(-1)^m(2m-1)! R\over (2k)^{2m-1}}$$ (16)

$$L_{-k}(-2k) = {(-1)^mR'(2m)!\over (2k)^{2m}},$$ (17)

where $R$ and $R'$ are Rational Numbers. $L_{+k}(1)$ can be expressed in terms of transcendentals by

$$L_d(1)=h(d) \kappa(d),$$ (18)

where $h(d)$ is the Class Number and $\kappa(d)$ is the Dirichlet Structure Constant. Some specific values of primitive $L$-series are

$$L_{-15}(1) = {2\pi\over\sqrt{15}}$$

$$L_{-11}(1) = {\pi\over\sqrt{11}}$$

$$L_{-8}(1) = {\pi\over 2\sqrt{2}}$$

$$L_{-7}(1) = {\pi\over\sqrt{7}}$$

$$L_{-4}(1) = {\textstyle{1\over 4}}\pi$$

$$L_{-3}(1) = {\pi\over 3\sqrt{3}}$$

$$L_{+5}(1) = {2\over\sqrt{5}} \ln\left({1+\sqrt{5}\over 2}\right)$$

$$L_{+8}(1) = {\ln(1+\sqrt{2}\,)\over\sqrt{2}}$$

$$L_{+12}(1) = {\ln(2+\sqrt{3}\,)\over\sqrt{3}}$$

$$L_{+13}(1) = {2\over\sqrt{13}} \ln\left({3+\sqrt{13}\over 2}\right)$$

$$L_{+17}(1) = {2\over\sqrt{17}} \ln(4+\sqrt{17}\,)$$

$$L_{+21}(1) = {2\over\sqrt{21}} \ln\left({5+\sqrt{21}\over 2}\right)$$

$$L_{+24}(1) = {\ln(5+2\sqrt{6}\,)\over\sqrt{6}}.$$

No general forms are known for $L_{-k}(2m)$ and $L_{+k}(2m-1)$ in terms of known transcendentals. For example,

$$L_{-4}(2) = \beta(2)\equiv K,$$ (19)

where $K$ is defined as Catalan's Constant.

See also Dirichlet Beta Function, Dirichlet Eta Function

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 $L$-Functions of Quadratic Fields.'' Math. Comput. 139, 786-796, 1977.

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.

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 $L$-Series.'' J. Phys. A: Math. Gen. 9, 1207-1214, 1976.