Dirichletova funkcija lambda
λ
(
s
)
{\displaystyle \lambda (s)\,}
je v matematiki specialna funkcija definirana kot Dirichletova L-vsota:[1] [2] [3]
λ
(
s
)
=
∑
n
=
0
∞
1
(
2
n
+
1
)
s
=
(
1
−
1
2
s
)
ζ
(
s
)
,
(
ℜ
(
s
)
>
1
)
,
{\displaystyle \lambda (s)=\sum _{n=0}^{\infty }{\frac {1}{(2n+1)^{s}=\left(1-{\frac {1}{2^{s}\right)\zeta (s),\qquad (\Re (s)>1)\!\,,}
kjer je
ζ
(
s
)
{\displaystyle \zeta (s)\,}
Riemannova funkcija ζ . Imenuje se po nemškem matematiku Petru Gustavu Lejeuneu Dirichletu .
Značilnosti
Dirichletova funkcija
λ
{\displaystyle \lambda \,}
je povezana z Riemannovo funkcijo ζ in Dirichletovo funkcijo η kot:
λ
(
s
)
2
s
−
1
=
ζ
(
s
)
2
s
=
η
(
s
)
2
s
−
2
{\displaystyle {\frac {\lambda (s)}{2^{s}-1}={\frac {\zeta (s)}{2^{s}={\frac {\eta (s)}{2^{s}-2}\!\,}
in:
2
λ
(
s
)
=
ζ
(
s
)
+
η
(
s
)
.
{\displaystyle 2\lambda (s)=\zeta (s)+\eta (s)\!\,.}
Dirichletova funkcija
λ
{\displaystyle \lambda \,}
je posebni primer Legendrove funkcije
χ
ν
(
z
)
{\displaystyle \chi _{\nu }(z)\,}
za
z
=
1
{\displaystyle z=1\,}
:
χ
n
(
1
)
=
λ
(
n
)
=
(
1
−
1
2
n
)
ζ
(
n
)
=
(
1
+
1
2
n
−
2
)
η
(
n
)
.
{\displaystyle \chi _{n}(1)=\lambda (n)=\left(1-{\frac {1}{2^{n}\right)\zeta (n)=\left(1+{\frac {1}{2^{n}-2}\right)\eta (n)\!\,.}
Posebne vrednosti Dirichletove funkcije λ
Za soda pozitivna cela števila velja:[2]
λ
(
2
n
)
=
(
2
2
n
−
1
)
(
−
1
)
n
−
1
B
2
n
π
2
n
2
(
2
n
)
!
,
(
n
∈
N
)
,
{\displaystyle \lambda (2n)=\left(2^{2n}-1\right){\frac {(-1)^{n-1}B_{2n}\pi ^{2n}{2(2n)!},\qquad (n\in \mathbb {N} )\!\,,}
kjer so B 2n Bernoullijeva števila .
Za liha pozitivna cela števila velja:[2]
λ
(
2
n
+
1
)
=
∑
k
=
1
n
(
(
−
1
)
k
−
1
λ
(
2
n
−
2
k
+
2
)
)
+
(
−
1
)
n
β
(
1
)
J
(
2
n
)
,
(
n
∈
N
)
,
{\displaystyle \lambda (2n+1)=\sum _{k=1}^{n}\left((-1)^{k-1}\lambda (2n-2k+2)\right)+(-1)^{n}\beta (1)J(2n),\qquad (n\in \mathbb {N} )\!\,,}
kjer je
β
(
s
)
{\displaystyle \beta (s)\,}
Dirichletova funkcija β ,
J
(
s
)
{\displaystyle J(s)\,}
pa integralska funkcija:[2]
J
(
s
)
=
1
Γ
(
s
+
1
)
2
π
∫
0
π
/
2
x
s
sin
x
d
x
,
{\displaystyle J(s)={\frac {1}{\Gamma (s+1)}{\frac {2}{\pi }\int _{0}^{\pi /2}{\frac {x^{s}{\sin x}\mathrm {d} x\!\,,}
kjer je
Γ
(
s
)
{\displaystyle \Gamma (s)\,}
funkcija Γ .
λ
(
−
3
)
=
−
7
ζ
(
−
3
)
=
−
7
120
=
−
0
,
058
3
¯
.
{\displaystyle \lambda (-3)=-7\zeta (-3)=-{\frac {7}{120}=-0,058{\overline {3}\!\,.}
λ
(
−
1
)
=
−
ζ
(
−
1
)
=
1
12
=
0
,
08
3
¯
.
{\displaystyle \lambda (-1)=-\zeta (-1)={\frac {1}{12}=0,08{\overline {3}\!\,.}
λ
(
0
)
=
0
.
{\displaystyle \lambda (0)=0\!\,.}
λ
(
1
/
2
)
=
(
1
−
2
2
)
ζ
(
1
/
2
)
=
−
0
,
427727932693
…
.
{\displaystyle \lambda (1/2)=\left(1-{\frac {\sqrt {2}{2}\right)\zeta (1/2)=-0,427727932693\ldots \!\,.}
λ
(
1
)
=
1
+
1
3
+
1
5
+
⋯
=
∞
.
{\displaystyle \lambda (1)=1+{\frac {1}{3}+{\frac {1}{5}+\cdots =\infty \!\,.}
λ
(
3
/
2
)
=
1
,
688761186655
…
.
{\displaystyle \lambda (3/2)=1,688761186655\ldots \!\,.}
λ
(
2
)
=
1
+
1
3
2
+
1
5
2
+
⋯
=
π
2
8
=
1
,
233700550136
…
,
{\displaystyle \lambda (2)=1+{\frac {1}{3^{2}+{\frac {1}{5^{2}+\cdots ={\frac {\pi ^{2}{8}=1,233700550136\ldots \!\,,}
(OEIS A111003 ).
λ
(
3
)
=
λ
(
2
)
J
(
1
)
−
β
(
1
)
J
(
2
)
=
7
ζ
(
3
)
8
=
1
,
051799790264
…
.
{\displaystyle \lambda (3)=\lambda (2)J(1)-\beta (1)J(2)={\frac {7\zeta (3)}{8}=1,051799790264\ldots \!\,.}
λ
(
4
)
=
1
+
1
3
4
+
1
5
4
+
⋯
=
π
4
96
=
1
,
014678031604
…
.
{\displaystyle \lambda (4)=1+{\frac {1}{3^{4}+{\frac {1}{5^{4}+\cdots ={\frac {\pi ^{4}{96}=1,014678031604\ldots \!\,.}
λ
(
5
)
=
λ
(
4
)
J
(
1
)
−
λ
(
2
)
J
(
3
)
+
β
(
1
)
J
(
4
)
=
31
ζ
(
5
)
32
=
1
,
004523762795
…
.
{\displaystyle \lambda (5)=\lambda (4)J(1)-\lambda (2)J(3)+\beta (1)J(4)={\frac {31\zeta (5)}{32}=1,004523762795\ldots \!\,.}
λ
(
6
)
=
1
+
1
3
6
+
1
5
6
+
⋯
=
π
6
960
=
1
,
001447076640
…
.
{\displaystyle \lambda (6)=1+{\frac {1}{3^{6}+{\frac {1}{5^{6}+\cdots ={\frac {\pi ^{6}{960}=1,001447076640\ldots \!\,.}
λ
(
7
)
=
λ
(
6
)
J
(
1
)
−
λ
(
4
)
J
(
3
)
+
λ
(
2
)
J
(
5
)
−
β
(
1
)
J
(
6
)
=
127
ζ
(
7
)
128
=
1
,
000471548652
…
.
{\displaystyle \lambda (7)=\lambda (6)J(1)-\lambda (4)J(3)+\lambda (2)J(5)-\beta (1)J(6)={\frac {127\zeta (7)}{128}=1,000471548652\ldots \!\,.}
Velja tudi zveza:[4]
log
D
=
∑
n
=
1
∞
(
−
1
)
2
n
+
1
n
(
2
D
π
)
2
n
λ
(
2
n
)
,
{\displaystyle \log D=\sum _{n=1}^{\infty }{\frac {(-1)^{2n+1}{n}\left({\frac {2D}{\pi }\right)^{2n}\lambda (2n)\!\,,}
kjer je
D
{\displaystyle D\,}
Dottiejino število .
Glej tudi
Sklici
Viri
Abramowitz, Milton; Stegun, Irene Anne (1972). »Riemann Zeta Function and other Sums of Recirocal Powers. §23.2«. Handbook of Mathematical Functions with Formulas, Graphs, and Mathematical Tables (9. izd.). New York: Dover. str. 807–808. ISBN 978-0486612720 . MR 0167642 . Glej razdelek §23.2
Cvijović, Djurdje (2007). »Integral representations of the Legendre chi function«. J. Math. Anal. Appl. Zv. 332. str. 1056–1062. arXiv :0911.4731 . doi :10.1016/j.jmaa.2006.10.083 .
Kim, JeonWon (2014). »Functional Equations related to the Dirichlet lambda and beta functions«. arXiv :1404.5467 [math:NT ].
Maresh, Owen (11. februar 2011). »the logarithm of the dottie number and Dirichlet's lambda function« . tsungfruve.wordpress.com (v angleščini). Pridobljeno 13. junija 2015 .
Spanier, J.; Oldham, K. B. (1987). An Atlas of Functions . New York: Hemisphere.
Zunanje povezave