§25.15 Dirichlet $\mathop{L\/}\nolimits$ -functions

Referenced by:: §24.17(iii)
Permalink:: http://dlmf.nist.gov/25.15

§25.15(i) Definitions and Basic Properties
§25.15(ii) Zeros

§25.15(i) Definitions and Basic Properties

Notes:: See Apostol (1976, pp. 231, 249, and 262–263).
Keywords:: Dirichlet -functions, infinite products
Permalink:: http://dlmf.nist.gov/25.15.i

The notation $\mathop{L\/}\nolimits\!\left(s,\chi\right)$ was introduced by Dirichlet (1837) for the meromorphic continuation of the function defined by the series

25.15.1 $\mathop{L\/}\nolimits\!\left(s,\chi\right)=\sum_{{n=1}}^{\infty}\frac{\chi(n)}% {n^{s}},$ $\realpart{s}>1$ ,

Defines:: $\mathop{L\/}\nolimits\!\left(s,\chi\right)$ : Dirichlet $\mathop{L\/}\nolimits$ -function
Symbols:: $\realpart{}$ : real part, : nonnegative integer, : complex variable and $\chi(n)$ : Dirichlet character
Permalink:: http://dlmf.nist.gov/25.15.E1
Encodings:: TeX, pMML, png

where $\chi(n)$ is a Dirichlet character $\;\;(\mathop{{\rm mod}}k)$ (§27.8). For the principal character $\chi_{1}\;\;(\mathop{{\rm mod}}k)$ , $\mathop{L\/}\nolimits\!\left(s,\chi_{1}\right)$ is analytic everywhere except for a simple pole at with residue $\mathop{\phi\/}\nolimits\!\left(k\right)/k$ , where $\mathop{\phi\/}\nolimits\!\left(k\right)$ is Euler’s totient function (§27.2). If $\chi\neq\chi_{1}$ , then $\mathop{L\/}\nolimits\!\left(s,\chi\right)$ is an entire function of .

25.15.2 $\mathop{L\/}\nolimits\!\left(s,\chi\right)=\prod_{p}\left(1-\frac{\chi(p)}{p^{% s}}\right)^{{-1}},$ $\realpart{s}>1$ ,

Symbols:: $\mathop{L\/}\nolimits\!\left(s,\chi\right)$ : Dirichlet $\mathop{L\/}\nolimits$ -function, $\realpart{}$ : real part, : prime number, : complex variable and $\chi(n)$ : Dirichlet character
Permalink:: http://dlmf.nist.gov/25.15.E2
Encodings:: TeX, pMML, png

with the product taken over all primes , beginning with . This implies that $\mathop{L\/}\nolimits\!\left(s,\chi\right)\neq 0$ if $\realpart{s}>1$ .

Equations (25.15.3) and (25.15.4) hold for all if $\chi\neq\chi_{1}$ , and for all ( $\neq 1$ ) if $\chi=\chi_{1}$ :

25.15.3 $\mathop{L\/}\nolimits\!\left(s,\chi\right)=k^{{-s}}\sum_{{r=1}}^{{k-1}}\chi(r)% \mathop{\zeta\/}\nolimits\!\left(s,\frac{r}{k}\right),$

Symbols:: $\mathop{L\/}\nolimits\!\left(s,\chi\right)$ : Dirichlet $\mathop{L\/}\nolimits$ -function, $\mathop{\zeta\/}\nolimits\!\left(s,a\right)$ : Hurwitz zeta function, : nonnegative integer, : complex variable and $\chi(n)$ : Dirichlet character
Referenced by:: §25.15(i)
Permalink:: http://dlmf.nist.gov/25.15.E3
Encodings:: TeX, pMML, png

25.15.4 $\mathop{L\/}\nolimits\!\left(s,\chi\right)=\mathop{L\/}\nolimits\!\left(s,\chi% _{0}\right)\prod_{{p\divides k}}\left(1-\frac{\chi_{0}(p)}{p^{s}}\right),$

Symbols:: $\mathop{L\/}\nolimits\!\left(s,\chi\right)$ : Dirichlet $\mathop{L\/}\nolimits$ -function, : nonnegative integer, : prime number, : complex variable and $\chi(n)$ : Dirichlet character
Referenced by:: §25.15(i)
Permalink:: http://dlmf.nist.gov/25.15.E4
Encodings:: TeX, pMML, png

where $\chi_{0}$ is a primitive character (mod ) for some positive divisor of (§27.8).

When $\chi$ is a primitive character (mod ) the $\mathop{L\/}\nolimits$ -functions satisfy the functional equation:

25.15.5 $\mathop{L\/}\nolimits\!\left(1-s,\chi\right)=\frac{k^{{s-1}}\mathop{\Gamma\/}% \nolimits\!\left(s\right)}{(2\pi)^{s}}\*{\left(e^{{-\pi is/2}}+\chi(-1)e^{{\pi is% /2}}\right)}\*G(\chi)\mathop{L\/}\nolimits\!\left(s,\overline{\chi}\right),$

Symbols:: $\mathop{L\/}\nolimits\!\left(s,\chi\right)$ : Dirichlet $\mathop{L\/}\nolimits$ -function, $\mathop{\Gamma\/}\nolimits\!\left(z\right)$ : gamma function, : base of exponential function, : nonnegative integer, : complex variable and $\chi(n)$ : Dirichlet character
Referenced by:: §25.15(ii)
Permalink:: http://dlmf.nist.gov/25.15.E5
Encodings:: TeX, pMML, png

where $\overline{\chi}$ is the complex conjugate of $\chi$ , and

25.15.6 $G(\chi)=\sum_{{r=1}}^{k}\chi(r)e^{{2\pi ir/k}}.$

Symbols:: : base of exponential function, : nonnegative integer and $\chi(n)$ : Dirichlet character
Permalink:: http://dlmf.nist.gov/25.15.E6
Encodings:: TeX, pMML, png

§25.15(ii) Zeros

Notes:: See Apostol (1985b) and Apostol (1976, pp. 142, 149, 268).
Keywords:: Dirichlet -functions, Dirichlet’s theorem, prime numbers
Permalink:: http://dlmf.nist.gov/25.15.ii

Since $\mathop{L\/}\nolimits\!\left(s,\chi\right)\neq 0$ if $\realpart{s}>1$ , (25.15.5) shows that for a primitive character $\chi$ the only zeros of $\mathop{L\/}\nolimits\!\left(s,\chi\right)$ for $\realpart{s}<0$ (the so-called trivial zeros) are as follows:

25.15.7 $\mathop{L\/}\nolimits\!\left(-2n,\chi\right)=0\text{ if }\chi(-1)=1,$ $n=0,1,2,\dots$ ,

Symbols:: $\mathop{L\/}\nolimits\!\left(s,\chi\right)$ : Dirichlet $\mathop{L\/}\nolimits$ -function, : nonnegative integer and $\chi(n)$ : Dirichlet character
Permalink:: http://dlmf.nist.gov/25.15.E7
Encodings:: TeX, pMML, png

25.15.8 $\mathop{L\/}\nolimits\!\left(-2n-1,\chi\right)=0\text{ if }\chi(-1)=-1,$ $n=0,1,2,\dots$ .

Symbols:: $\mathop{L\/}\nolimits\!\left(s,\chi\right)$ : Dirichlet $\mathop{L\/}\nolimits$ -function, : nonnegative integer and $\chi(n)$ : Dirichlet character
Permalink:: http://dlmf.nist.gov/25.15.E8
Encodings:: TeX, pMML, png

There are also infinitely many zeros in the critical strip $0\leq\realpart{s}\leq 1$ , located symmetrically about the critical line $\realpart{s}=\frac{1}{2}$ , but not necessarily symmetrically about the real axis.

25.15.9 $\mathop{L\/}\nolimits\!\left(1,\chi\right)\neq 0\text{ if }\chi\neq\chi_{1},$

Symbols:: $\mathop{L\/}\nolimits\!\left(s,\chi\right)$ : Dirichlet $\mathop{L\/}\nolimits$ -function and $\chi(n)$ : Dirichlet character
Permalink:: http://dlmf.nist.gov/25.15.E9
Encodings:: TeX, pMML, png

where $\chi_{1}$ is the principal character $\;\;(\mathop{{\rm mod}}k)$ . This result plays an important role in the proof of Dirichlet’s theorem on primes in arithmetic progressions (§27.11). Related results are:

25.15.10 $\mathop{L\/}\nolimits\!\left(0,\chi\right)=\begin{cases}\displaystyle-\frac{1}% {k}\sum_{{r=1}}^{k}r\chi(r),&\chi\neq\chi_{1},\\ 0,&\chi=\chi_{1}.\end{cases}$

Symbols:: $\mathop{L\/}\nolimits\!\left(s,\chi\right)$ : Dirichlet $\mathop{L\/}\nolimits$ -function, : nonnegative integer and $\chi(n)$ : Dirichlet character
Permalink:: http://dlmf.nist.gov/25.15.E10
Encodings:: TeX, pMML, png

© 2010–2013 NIST / Privacy Policy / Disclaimer / Feedback; Version 1.0.6; Release date 2013-05-06 Math-Image version (See MathML) . A Printed Companionis available. 25.14 Lerch’s Transcendent 25.16 Mathematical Applications

§25.15 Dirichlet -functions

Contents

§25.15(i) Definitions and Basic Properties

§25.15(ii) Zeros

§25.15 Dirichlet $\mathop{L\/}\nolimits$ -functions