Dirichlet L-function

Since a Dirichlet character $\backslash chi$ is completely multiplicative, its $L$-function can also be written as an Euler product in the half-plane of absolute convergence:

:$L(s,\backslash chi)=\backslash prod\_p\backslash left(1-\backslash chi(p)p^\{-s\}\backslash right)^\{-1\}\backslash text\{\; for\; \}\backslash text\{Re\}(s)\; >\; 1,$

where the product is over all prime numbers.{{harvnb|Apostol|1976|loc=Theorem 11.7}}

Results about L-functions are often stated more simply if the character is assumed to be primitive, although the results typically can be extended to imprimitive characters with minor complications.{{harvnb|Davenport|2000|loc=chapter 5}} This is because of the relationship between a imprimitive character $\backslash chi$ and the primitive character $\backslash chi^\backslash star$ which induces it:{{harvnb|Davenport|2000|loc=chapter 5, equation (2)}}

:$$

\chi(n) =

\begin{cases}

\chi^\star(n), & \mathrm{if} \gcd(n,q) = 1 \\

0, & \mathrm{if} \gcd(n,q) \ne 1

\end{cases}

(Here, q is the modulus of χ.) An application of the Euler product gives a simple relationship between the corresponding L-functions:{{harvnb|Davenport|2000|loc=chapter 5, equation (3)}}{{harvnb|Montgomery|Vaughan|2006|p=282}}

:$$

L(s,\chi) = L(s,\chi^\star) \prod_{p \,|\, q}\left(1 - \frac{\chi^\star(p)}{p^s} \right)

(This formula holds for all s, by analytic continuation, even though the Euler product is only valid when Re(s) > 1.) The formula shows that the L-function of χ is equal to the L-function of the primitive character which induces χ, multiplied by only a finite number of factors.{{harvnb|Apostol|1976|p=262}}

As a special case, the L-function of the principal character $\backslash chi\_0$ modulo q can be expressed in terms of the Riemann zeta function:{{harvnb|Ireland|Rosen|1990|loc=chapter 16, section 4}}{{harvnb|Montgomery|Vaughan|2006|p=121}}

:$$

L(s,\chi_0) = \zeta(s) \prod_{p \,|\, q}(1 - p^{-s})

Dirichlet L-functions satisfy a functional equation, which provides a way to analytically continue them throughout the complex plane. The functional equation relates the value of $L(s,\backslash chi)$ to the value of $L(1-s,\; \backslash overline\{\backslash chi\})$. Let χ be a primitive character modulo q, where q > 1. One way to express the functional equation is:

:$L(s,\backslash chi)\; =\; W(\backslash chi)\; 2^s\; \backslash pi^\{s-1\}\; q^\{1/2-s\}\; \backslash sin\; \backslash left(\; \backslash frac\{\backslash pi\}\{2\}\; (s\; +\; \backslash delta)\; \backslash right)\; \backslash Gamma(1-s)\; L(1-s,\; \backslash overline\{\backslash chi\}).$

In this equation, Γ denotes the gamma function;

:$\backslash chi(-1)=(-1)^\{\backslash delta\}$ ; and

:$W(\backslash chi)\; =\; \backslash frac\{\backslash tau(\backslash chi)\}\{i^\{\backslash delta\}\; \backslash sqrt\{q\}\}$

where τ{{hairsp}}({{hairsp}}χ) is a Gauss sum:

:$\backslash tau(\backslash chi)\; =\; \backslash sum\_\{a=1\}^q\; \backslash chi(a)\backslash exp(2\backslash pi\; ia/q).$

It is a property of Gauss sums that |τ{{hairsp}}({{hairsp}}χ){{hairsp}}| = q^1/2, so |W{{hairsp}}({{hairsp}}χ){{hairsp}}| = 1.{{harvnb|Montgomery|Vaughan|2006|p=332}}{{harvnb|Iwaniec|Kowalski|2004|p=84}}

Another way to state the functional equation is in terms of

:$\backslash Lambda(s,\backslash chi)\; =\; q\; ^\{s/2\}\; \backslash pi^\{-(s+\backslash delta)/2\}\; \backslash operatorname\{\backslash Gamma\}\backslash left(\backslash frac\{s+\backslash delta\}\{2\}\backslash right)\; L(s,\backslash chi).$

The functional equation can be expressed as:

:$\backslash Lambda(s,\backslash chi)\; =\; W(\backslash chi)\; \backslash Lambda(1-s,\backslash overline\{\backslash chi\}).$

The functional equation implies that $L(s,\backslash chi)$ (and $\backslash Lambda(s,\backslash chi)$) are entire functions of s. (Again, this assumes that χ is primitive character modulo q with q > 1. If q = 1, then $L(s,\backslash chi)\; =\; \backslash zeta(s)$ has a pole at s = 1.){{harvnb|Montgomery|Vaughan|2006|p=333}}

For generalizations, see: Functional equation (L-function).

Image:Mplwp dirichlet beta.svg), with trivial zeros at the negative odd integers]]

Let χ be a primitive character modulo q, with q > 1.

There are no zeros of L(s, χ) with Re(s) > 1. For Re(s) < 0, there are zeros at certain negative integers s:

• If χ(−1) = 1, the only zeros of L(s, χ) with Re(s) < 0 are simple zeros at −2, −4, −6, .... (There is also a zero at s = 0.) These correspond to the poles of $\backslash textstyle\; \backslash Gamma(\backslash frac\{s\}\{2\})$.{{harvnb|Davenport|2000|loc=chapter 9}}
• If χ(−1) = −1, then the only zeros of L(s, χ) with Re(s) < 0 are simple zeros at −1, −3, −5, .... These correspond to the poles of $\backslash textstyle\; \backslash Gamma(\backslash frac\{s+1\}\{2\})$.

These are called the trivial zeros.

The remaining zeros lie in the critical strip 0 ≤ Re(s) ≤ 1, and are called the non-trivial zeros. The non-trivial zeros are symmetrical about the critical line Re(s) = 1/2. That is, if $L(\backslash rho,\backslash chi)=0$ then $L(1-\backslash overline\{\backslash rho\},\backslash chi)=0$ too, because of the functional equation. If χ is a real character, then the non-trivial zeros are also symmetrical about the real axis, but not if χ is a complex character. The generalized Riemann hypothesis is the conjecture that all the non-trivial zeros lie on the critical line Re(s) = 1/2.

Up to the possible existence of a Siegel zero, zero-free regions including and beyond the line Re(s) = 1 similar to that of the Riemann zeta function are known to exist for all Dirichlet L-functions: for example, for χ a non-real character of modulus q, we have

:

for β + iγ a non-real zero.{{cite book |last=Montgomery |first=Hugh L. |author-link=Hugh Montgomery (mathematician) |title=Ten lectures on the interface between analytic number theory and harmonic analysis |series=Regional Conference Series in Mathematics |volume=84 |location=Providence, RI |publisher=American Mathematical Society |year=1994 |isbn=0-8218-0737-4 |zbl=0814.11001 |page=163}}

The Dirichlet L-functions may be written as a linear combination of the Hurwitz zeta function at rational values. Fixing an integer k ≥ 1, the Dirichlet L-functions for characters modulo k are linear combinations, with constant coefficients, of the ζ(s,a) where a = r/k and r = 1, 2, ..., k. This means that the Hurwitz zeta function for rational a has analytic properties that are closely related to the Dirichlet L-functions. Specifically, let χ be a character modulo k. Then we can write its Dirichlet L-function as:{{harvnb|Apostol|1976|p=249}}

:$L(s,\backslash chi)\; =\; \backslash sum\_\{n=1\}^\backslash infty\; \backslash frac\{\backslash chi(n)\}\{n^s\}$

= \frac{1}{k^s} \sum_{r=1}^k \chi(r) \operatorname{\zeta}\left(s,\frac{r}{k}\right).

• Generalized Riemann hypothesis
• L-function
• Modularity theorem
• Artin conjecture
• Special values of L-functions

{{reflist}}

• {{Apostol IANT}}
• {{dlmf|id=25.15|first=T. M.|last=Apostol}}
• {{cite book|first=H.|last=Davenport|author-link=Harold Davenport

|title=Multiplicative Number Theory

|publisher=Springer

|year=2000

|edition=3rd

|isbn=0-387-95097-4}}

• {{Cite journal

| last=Dirichlet

| first=P. G. L.

| author-link=Peter Gustav Lejeune Dirichlet

| title=Beweis des Satzes, dass jede unbegrenzte arithmetische Progression, deren erstes Glied und Differenz ganze Zahlen ohne gemeinschaftlichen Factor sind, unendlich viele Primzahlen enthält

| journal=Abhand. Ak. Wiss. Berlin

| volume=48

| year=1837

}}

• {{cite book|first1=Kenneth|last1=Ireland|first2=Michael|last2=Rosen|author-link2=Michael Rosen (mathematician)|title=A Classical Introduction to Modern Number Theory|edition=2nd|publisher=Springer-Verlag|year=1990}}
• {{cite book|first1=Hugh L.|last1=Montgomery |author-link=Hugh Montgomery (mathematician)|first2=Robert C.|last2=Vaughan |author-link2=Robert Charles Vaughan (mathematician) | title=Multiplicative number theory. I. Classical theory| series=Cambridge tracts in advanced mathematics| volume=97| publisher=Cambridge University Press|year=2006| isbn=978-0-521-84903-6}}
• {{cite book

|last1=Iwaniec

|first1=Henryk

|author-link=Henryk Iwaniec

|last2=Kowalski

|first2=Emmanuel

|year=2004

|title=Analytic Number Theory

|series=American Mathematical Society Colloquium Publications

|volume=53

|location=Providence, RI

|publisher=American Mathematical Society

}}

• {{springer|title=Dirichlet-L-function|id=p/d032890}}

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Category:Zeta and L-functions