Dirichlet L-function

In mathematics, a Dirichlet L-series is a function of the form

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

where ${\displaystyle \chi }$ is a Dirichlet character and ${\displaystyle s}$ a complex variable with real part greater than ${\displaystyle 1}$. 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; it is then called a Dirichlet L-function.

These functions are named after Peter Gustav Lejeune Dirichlet who introduced them in 1837^[1] to prove his theorem on primes in arithmetic progressions. In his proof, Dirichlet showed that ${\displaystyle L(s,\chi )}$ is non-zero at ${\displaystyle s=1}$. Moreover, if ${\displaystyle \chi }$ is principal, then the corresponding Dirichlet L-function has a simple pole at ${\displaystyle s=1}$. Otherwise, the L-function is entire.

Euler product

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

${\displaystyle L(s,\chi )=\prod _{p}\left(1-\chi (p)p^{-s}\right)^{-1}{\text{ for }}{\text{Re}}(s)>1,}$

where the product is over all prime numbers.^[2]

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.^[3] This is because of the relationship between a imprimitive character ${\displaystyle \chi }$ and the primitive character ${\displaystyle \chi ^{\star }}$ which induces it:^[4]

${\displaystyle \chi (n)={\begin{cases}\chi ^{\star }(n)&\mathrm {if} \gcd(n,q)=1,\\\;\;\;0&\mathrm {otherwise} .\end{cases}}}$

(Here, ${\displaystyle q}$ is the modulus of ${\displaystyle \chi }$.) An application of the Euler product gives a simple relationship between the corresponding L-functions:^[5][6]

${\displaystyle L(s,\chi )=L(s,\chi ^{\star })\prod _{p\,|\,q}\left(1-{\frac {\chi ^{\star }(p)}{p^{s}}}\right).}$

By analytic continuation, this formula holds for all complex ${\displaystyle s}$, even though the Euler product is only valid when ${\displaystyle \operatorname {Re} (s)>1}$. The formula shows that the L-function of ${\displaystyle \chi }$ is equal to the L-function of the primitive character which induces ${\displaystyle \chi }$, multiplied by only a finite number of factors.^[7]

As a special case, the L-function of the principal character ${\displaystyle \chi _{0}}$ modulo ${\displaystyle q}$ can be expressed in terms of the Riemann zeta function:^[8][9]

${\displaystyle L(s,\chi _{0})=\zeta (s)\prod _{p\,|\,q}(1-p^{-s}).}$

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 values of ${\displaystyle L(s,\chi )}$ to the values of ${\displaystyle L(1-s,{\overline {\chi }})}$.

Let ${\displaystyle \chi }$ be a primitive character modulo ${\displaystyle q}$, where ${\displaystyle q>1}$. One way to express the functional equation is as^[10]

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

where ${\displaystyle \Gamma }$ is the gamma function, ${\displaystyle \chi (-1)=(-1)^{\delta }}$, and

${\displaystyle W(\chi )={\frac {\tau (\chi )}{i^{\delta }{\sqrt {q}}}},}$

where ${\displaystyle \tau (\chi )}$ is the Gauss sum

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

It is a property of Gauss sums that ${\displaystyle |\tau (\chi )|={\sqrt {q}}}$, so ${\displaystyle |W(\chi )|=1}$.^[11][12] Another functional equation is

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

which can be expressed as^[10][12]

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

This implies that ${\displaystyle L(s,\chi )}$ and ${\displaystyle \Lambda (s,\chi )}$ are entire functions of ${\displaystyle s}$. Again, this assumes that ${\displaystyle \chi }$ is primitive character modulo ${\displaystyle q}$ with ${\displaystyle q>1}$. If ${\displaystyle q=1}$, then ${\displaystyle L(s,\chi )=\zeta (s)}$ has a pole at ${\displaystyle s=1}$.^[10][12]

For generalizations, see the article on functional equations of L-functions.

Zeros

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 ${\displaystyle \chi }$ be a primitive character modulo ${\displaystyle q}$, with ${\displaystyle q>1}$.

There are no zeros of ${\displaystyle L(s,\chi )}$ with ${\displaystyle \operatorname {Re} (s)>1}$. For ${\displaystyle \operatorname {Re} (s)<0}$, there are zeros at certain negative integers ${\displaystyle s}$:

• If ${\displaystyle \chi (-1)=1}$, the only zeros of ${\displaystyle L(s,\chi )}$ with ${\displaystyle \operatorname {Re} (s)<0}$ are simple zeros at ${\displaystyle -2,-4,-6,\dots }$ There is also a zero at ${\displaystyle s=0}$ when ${\displaystyle \chi }$ is non-principal. These correspond to the poles of ${\displaystyle \textstyle \Gamma ({\frac {s}{2}})}$.^[13]
• If ${\displaystyle \chi (-1)=-1}$, then the only zeros of ${\displaystyle L(s,\chi )}$ with ${\displaystyle \operatorname {Re} (s)<0}$ are simple zeros at ${\displaystyle -1,-3,-5,\dots }$ These correspond to the poles of ${\displaystyle \textstyle \Gamma ({\frac {s+1}{2}})}$.^[13]

These are called the trivial zeros.^[10]

The remaining zeros lie in the critical strip ${\displaystyle 0\leq \operatorname {Re} (s)\leq 1}$, and are called the non-trivial zeros. The non-trivial zeros are symmetrical about the critical line ${\displaystyle \operatorname {Re} (s)=1/2}$. That is, if ${\displaystyle L(\rho ,\chi )=0}$, then ${\displaystyle L(1-{\overline {\rho }},\chi )=0}$ too because of the functional equation. If ${\displaystyle \chi }$ is a real character, then the non-trivial zeros are also symmetrical about the real axis, but not if ${\displaystyle \chi }$ is a complex character. The generalized Riemann hypothesis is the conjecture that all the non-trivial zeros lie on the critical line ${\displaystyle \operatorname {Re} (s)=1/2}$.^[10]

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

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

for ${\displaystyle \beta +i\gamma }$ a non-real zero.^[14]

Relation to the Hurwitz zeta function

Dirichlet L-functions may be written as linear combinations of the Hurwitz zeta function at rational values. Fixing an integer ${\displaystyle k\geq 1}$, Dirichlet L-functions for characters modulo ${\displaystyle k}$ are linear combinations with constant coefficients of the ${\displaystyle \zeta (s,a)}$ where ${\displaystyle a=r/k}$ and ${\displaystyle r=1,2,\dots ,k}$. This means that the Hurwitz zeta function for rational ${\displaystyle a}$ has analytic properties that are closely related to the Dirichlet L-functions. Specifically, if ${\displaystyle \chi }$ is a character modulo ${\displaystyle k}$, we can write its Dirichlet L-function as^[15]

${\displaystyle L(s,\chi )=\sum _{n=1}^{\infty }{\frac {\chi (n)}{n^{s}}}={\frac {1}{k^{s}}}\sum _{r=1}^{k}\chi (r)\operatorname {\zeta } \left(s,{\frac {r}{k}}\right).}$

See also

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