Dirichlet L-function

In mathematics, a Dirichlet <math>L</math>-series is a function of the form

<math>L(s,\chi) = \sum_{n=1}^\infty \frac{\chi(n)}{n^s}.</math>

where <math> \chi </math> is a Dirichlet character and <math> s </math> a complex variable with real part greater than <math> 1 </math>. It is a special case of a Dirichlet series. By analytic continuation, it can be extended to a meromorphic function on the whole complex plane, and is then called a Dirichlet <math> L </math>-function and also denoted <math> L ( s , \chi) </math>.

These functions are named after Peter Gustav Lejeune Dirichlet who introduced them in Template:Harv to prove the theorem on primes in arithmetic progressions that also bears his name. In the course of the proof, Dirichlet shows that <math> L ( s , \chi) </math> is non-zero at <math> s = 1 </math>. Moreover, if <math> \chi </math> is principal, then the corresponding Dirichlet <math> L </math>-function has a simple pole at <math> s = 1 </math>. Otherwise, the <math> L </math>-function is entire.

Euler product

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

<math>L(s,\chi)=\prod_p\left(1-\chi(p)p^{-s}\right)^{-1}\text{ for }\text{Re}(s) > 1,</math>

where the product is over all prime numbers.<ref>Template:Harvnb</ref>

Primitive characters

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.<ref>Template:Harvnb</ref> This is because of the relationship between a imprimitive character <math>\chi</math> and the primitive character <math>\chi^\star</math> which induces it:<ref>Template:Harvnb</ref>

<math>

 \chi(n) =
   \begin{cases}
     \chi^\star(n), & \mathrm{if} \gcd(n,q) = 1 \\
     0, & \mathrm{if} \gcd(n,q) \ne 1
   \end{cases}

</math> (Here, q is the modulus of χ.) An application of the Euler product gives a simple relationship between the corresponding L-functions:<ref>Template:Harvnb</ref><ref>Template:Harvnb</ref>

<math>

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

</math> (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.<ref>Template:Harvnb</ref>

As a special case, the L-function of the principal character <math>\chi_0</math> modulo q can be expressed in terms of the Riemann zeta function:<ref>Template:Harvnb</ref><ref>Template:Harvnb</ref>

<math>

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

</math>

Functional equation

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 <math>L(s,\chi)</math> to the value of <math>L(1-s, \overline{\chi})</math>. Let χ be a primitive character modulo q, where q > 1. One way to express the functional equation is:<ref name="MontgomeryVaughan333" />

<math>L(s,\chi) = W(\chi) 2^s \pi^{s-1} q^{1/2-s} \sin \left( \frac{\pi}{2} (s + \delta) \right) \Gamma(1-s) L(1-s, \overline{\chi}).</math>

In this equation, Γ denotes the gamma function;

<math>\chi(-1)=(-1)^{\delta}</math> ; and

<math>W(\chi) = \frac{\tau(\chi)}{i^{\delta} \sqrt{q}}</math>

where τTemplate:Hairsp(Template:Hairspχ) is a Gauss sum:

<math>\tau(\chi) = \sum_{a=1}^q \chi(a)\exp(2\pi ia/q).</math>

It is a property of Gauss sums that |τTemplate:Hairsp(Template:Hairspχ)Template:Hairsp| = q^1/2, so |WTemplate:Hairsp(Template:Hairspχ)Template:Hairsp| = 1.<ref name="MontgomeryVaughan332">Template:Harvnb</ref><ref name="IwaniecKowalski84">Template:Harvnb</ref>

Another way to state the functional equation is in terms of

<math>\Lambda(s,\chi) = q ^{s/2} \pi^{-(s+\delta)/2} \operatorname{\Gamma}\left(\frac{s+\delta}{2}\right) L(s,\chi).</math>

The functional equation can be expressed as:<ref name="MontgomeryVaughan333" /><ref name="IwaniecKowalski84" />

<math>\Lambda(s,\chi) = W(\chi) \Lambda(1-s,\overline{\chi}).</math>

The functional equation implies that <math>L(s,\chi)</math> (and <math>\Lambda(s,\chi)</math>) are entire functions of s. (Again, this assumes that χ is primitive character modulo q with q > 1. If q = 1, then <math>L(s,\chi) = \zeta(s)</math> has a pole at s = 1.)<ref name="MontgomeryVaughan333">Template:Harvnb</ref><ref name="IwaniecKowalski84" />

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

Zeros

File:Mplwp dirichlet beta.svg

The Dirichlet L-function L(s, χ) = 1 − 3^−s + 5^−s − 7^−s + ⋅⋅⋅ (sometimes given the special name Dirichlet beta function), 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 <math>\textstyle \Gamma(\frac{s}{2})</math>.<ref name="DavenportCh9">Template:Harvnb</ref>
• 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 <math>\textstyle \Gamma(\frac{s+1}{2})</math>.<ref name="DavenportCh9" />

These are called the trivial zeros.<ref name="MontgomeryVaughan333"/>

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 <math>L(\rho,\chi)=0</math> then <math>L(1-\overline{\rho},\chi)=0</math> 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.<ref name="MontgomeryVaughan333" />

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

<math> \beta < 1 - \frac{c}{\log\!\!\; \big(q(2+|\gamma|)\big)} \ </math>

for β + iγ a non-real zero.<ref>Template:Cite book</ref>

Relation to the Hurwitz zeta function

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:<ref>Template:Harvnb</ref>

<math>L(s,\chi) = \sum_{n=1}^\infty \frac{\chi(n)}{n^s}

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

See also

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

References

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