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.

Zeros of the Dirichlet L-functions

If χ is a primitive character with χ(−1) = 1, then the only zeros of L(s,χ) with Re(s) < 0 are at the negative even integers. If χ is a primitive character with χ(−1) = −1, then the only zeros of L(s,χ) with Re(s) < 0 are at the negative odd integers.

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.^[1]

Just as the Riemann zeta function is conjectured to obey the Riemann hypothesis, so the Dirichlet L-functions are conjectured to obey the generalized Riemann hypothesis.

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.^[2]

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^ak^{1/2}}{\tau(\chi)}\Lambda(s,\chi).

Here we wrote τ(χ) for the Gauss sum

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

Note that |τ(χ)| = k^1/2.

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).

In particular, the Dirichlet L-function of the trivial character (which implies the modulus k is prime) yields the Riemann zeta-function:

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

Notes

↑ Montgomery, Hugh L. (1994). Ten lectures on the interface between analytic number theory and harmonic analysis. Regional Conference Series in Mathematics 84. Providence, RI: American Mathematical Society. p. 163. ISBN 0-8218-0737-4. Zbl 0814.11001.
↑ Apostol 1976, Theorem 11.17

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., NIST Handbook of Mathematical Functions, Cambridge University Press, ISBN 978-0521192255, MR 2723248
H. Davenport (2000). Multiplicative Number Theory. 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.
Hazewinkel, Michiel, ed. (2001), "Dirichlet-L-function", Encyclopedia of Mathematics, Springer, ISBN 978-1-55608-010-4

L-functions in number theory

Analytic examples	Riemann zeta function Dirichlet L-functions L-functions of Hecke characters Automorphic L-functions Selberg class

Algebraic examples	Dedekind zeta functions Artin L-functions Hasse–Weil L-functions Motivic L-functions

Theorems	Analytic class number formula Riemann–von Mangoldt formula Weil conjectures

Analytic conjectures	Riemann hypothesis Generalized Riemann hypothesis Lindelöf hypothesis Ramanujan–Petersson conjecture Artin conjecture

Algebraic conjectures	Birch and Swinnerton-Dyer conjecture Deligne's conjecture Beilinson conjectures Bloch–Kato conjecture Langlands conjecture

p-adic L-functions	Main conjecture of Iwasawa theory Selmer group Euler system

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