In mathematics, a Dirichlet L-series is a function of the form
Here 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. Otherwise, the L-function is entire.
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 and the primitive character which induces it:
(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.
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 to the value of . Let χ be a primitive character modulo q, where q > 1. One way to express the functional equation is:
In this equation, Γ denotes the Gamma function; a is 0 if χ(−1) = 1, or 1 if χ(−1) = −1; and
where τ ( χ) is a Gauss sum:
Another way to state the functional equation is in terms of
For generalizations, see: Functional equation (L-function).
Let χ be a primitive character modulo q, with q > 1.
- 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 .
- 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 .
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 then 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.
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:
- Generalized Riemann hypothesis
- Modularity theorem
- Artin conjecture
- Special values of L-functions
- Apostol 1976, Theorem 11.7 harvnb error: no target: CITEREFApostol1976 (help)
- Davenport 2000, chapter 5
- Davenport 2000, chapter 5, equation (2)
- Davenport 2000, chapter 5, equation (3)
- Montgomery & Vaughan 2006, p. 282
- Apostol 1976, p. 262 harvnb error: no target: CITEREFApostol1976 (help)
- Ireland & Rosen 1990, chapter 16, section 4
- Montgomery & Vaughan 2006, p. 121
- Montgomery & Vaughan 2006, p. 333
- Montgomery & Vaughan 2006, p. 332
- Iwaniec & Kowalski, p. 84 harvnb error: no target: CITEREFIwaniecKowalski (help)
- Montgomery & Vaughan 2006, p. 333
- Davenport 2000, chapter 9
- Montgomery, Hugh L. (1994). Ten lectures on the interface between analytic number theory and harmonic analysis. Regional Conference Series in Mathematics. Vol. 84. Providence, RI: American Mathematical Society. p. 163. ISBN 0-8218-0737-4. Zbl 0814.11001.
- Apostol 1976, p. 249 harvnb error: no target: CITEREFApostol1976 (help)
- 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.
- 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.
- Iwaniec, Henryk; Kowalski, Emmanuel (2004). Analytic Number Theory. American Mathematical Society Colloquium Publications. Vol. 53. Providence, RI: American Mathematical Society.
- "Dirichlet-L-function", Encyclopedia of Mathematics, EMS Press, 2001