Dirichlet L-function

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

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

Here χ is a Dirichlet character and s a complex variable with real part greater than 1. By analytic continuation, this function can be extended to a meromorphic function on the whole complex plane, and is then called a Dirichlet L-function and also denoted L(s, χ).

These functions are named after Peter Gustav Lejeune Dirichlet who introduced them in (Dirichlet 1837) to prove the theorem on primes in arithmetic progressions that also bears his name. In the course of the proof, Dirichlet shows that L(s, χ) is non-zero at s = 1. Moreover, if χ is principal, then the corresponding Dirichlet L-function has a simple pole at s = 1.

Euler product

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

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

where the product is over all prime numbers.

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

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

(Here, k is the modulus of χ.) An application of the Euler product gives a simple relationship between the corresponding L-functions:

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

(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.

As a special case, the L-function of the principal character $\chi _{0}$ modulo k can be expressed in terms of the Riemann zeta function:

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

Functional equation

Let us assume that χ is a primitive character to the modulus k. Defining

$\Lambda (s,\chi )=\left({\frac {\pi }{k}}\right)^{-(s+a)/2}\Gamma \left({\frac {s+a}{2}}\right)L(s,\chi ),$

where Γ denotes the Gamma function and the symbol a is given by

$a={\begin{cases}0;&{\mbox{if }}\chi (-1)=1,\\1;&{\mbox{if }}\chi (-1)=-1,\end{cases}}$

one has the functional equation

$\Lambda (1-s,{\overline {\chi }})={\frac {i^{a}k^{1/2}}{\tau (\chi )}}\Lambda (s,\chi ),$

where τ(χ) is the Gauss sum

$\sum _{n=1}^{k}\chi (n)\exp(2\pi in/k).$

Note that |τ(χ)| = k^1/2.

Zeros

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 $\textstyle \Gamma ({\frac {s}{2}})$.
• 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 $\textstyle \Gamma ({\frac {s+1}{2}})$.

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 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

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

for β + iγ a non-real zero.

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,q) where q = m/k and m = 1, 2, ..., k. This means that the Hurwitz zeta-function for rational q 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

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

See also

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

References

• Apostol, Tom M. (1976), Introduction to analytic number theory, Undergraduate Texts in Mathematics, New York-Heidelberg: Springer-Verlag, ISBN 978-0-387-90163-3, MR 0434929, Zbl 0335.10001
• Apostol, T. M. (2010), "Dirichlet L-function", in Olver, Frank W. J.; Lozier, Daniel M.; Boisvert, Ronald F.; Clark, Charles W. (eds.), NIST Handbook of Mathematical Functions, Cambridge University Press, ISBN 978-0-521-19225-5, MR 2723248.
• Davenport, H. (2000). Multiplicative Number Theory (3rd ed.). Springer. ISBN 0-387-95097-4.
• Dirichlet, P. G. L. (1837). "Beweis des Satzes, dass jede unbegrenzte arithmetische Progression, deren erstes Glied und Differenz ganze Zahlen ohne gemeinschaftlichen Factor sind, unendlich viele Primzahlen enthält". Abhand. Ak. Wiss. Berlin. 48. {{cite journal}}: Invalid |ref=harv (help)CS1 maint: postscript (link)
• Ireland, Kenneth; Rosen, Michael (1990). A Classical Introduction to Modern Number Theory (2nd ed.). Springer-Verlag.
• Montgomery, Hugh L.; Vaughan, Robert C. (2006). Multiplicative number theory. I. Classical theory. Cambridge tracts in advanced mathematics. Vol. 97. Cambridge University Press. ISBN 978-0-521-84903-6.
• "Dirichlet-L-function", Encyclopedia of Mathematics, EMS Press, 2001 [1994]