“Dirichlet L-function”的意思、由来-开放百科全书

词条

Dirichlet L-function

释义

Zeros of the Dirichlet L-functions
Euler product
Functional equation
Relation to the Hurwitz zeta-function
See also
Notes
References

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

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 {{harv|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 {{Nowrap|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

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:

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

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

one has the functional equation

Here we wrote τ(χ) for the Gauss sum

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

Notes

1. ^{{cite book | last=Montgomery | first=Hugh L. | author-link=Hugh Montgomery (mathematician) | title=Ten lectures on the interface between analytic number theory and harmonic analysis | series=Regional Conference Series in Mathematics | volume=84 | location=Providence, RI | publisher=American Mathematical Society | year=1994 | isbn=0-8218-0737-4 | zbl=0814.11001 | page=163 }}
2. ^{{harvnb|Apostol|1976|loc=Theorem 11.7}}

References

{{Apostol IANT}}
{{dlmf|id=25.15|first=T. M.|last=Apostol}}
{{cite book|author=H. Davenport

| title=Multiplicative Number Theory
|publisher=Springer
|year=2000
|isbn=0-387-95097-4}}

{{Cite journal

| last=Dirichlet
| first=P. G. L.
| author-link=Peter Gustav Lejeune Dirichlet
| title=Beweis des Satzes, dass jede unbegrenzte arithmetische Progression, deren erstes Glied und Differenz ganze Zahlen ohne gemeinschaftlichen Factor sind, unendlich viele Primzahlen enthält
| journal=Abhand. Ak. Wiss. Berlin
| volume=48
| year=1837
| ref=harv
| postscript=.
}}

{{springer|title=Dirichlet-L-function|id=p/d032890}}

1 : Zeta and L-functions

随便看