Particular values of the Riemann zeta function
In mathematics , the Riemann zeta function is a function in complex analysis , which is also important in number theory . It is often denoted ζ (s )Bernhard Riemann . When the argument s is a real number greater than one, the zeta function satisfies the equation
ζ
(
s
)
=
∑
n
=
1
∞
1
n
s
.
{\displaystyle \zeta (s)=\sum _{n=1}^{\infty }{\frac {1}{n^{s}}}\,.}
It can therefore provide the sum of various convergent
infinite series , such as
ζ
(
2
)
=
1
1
2
+
{\textstyle \zeta (2)={\frac {1}{1^{2}}}+}
1
2
2
+
{\textstyle {\frac {1}{2^{2}}}+}
1
3
2
+
…
.
{\textstyle {\frac {1}{3^{2}}}+\ldots \,.}
Explicit or numerically efficient formulae exist for
ζ (s ) at integer arguments, all of which have real values, including this example. This article lists these formulae, together with tables of values. It also includes derivatives and some series composed of the zeta function at integer arguments.
The same equation in s above also holds when s is a complex number whose real part is greater than one, ensuring that the infinite sum still converges. The zeta function can then be extended to the whole of the complex plane by analytic continuation , except for a simple pole at s = 1complex derivative exists in this more general region, making the zeta function a meromorphic function . The above equation no longer applies for these extended values of s , for which the corresponding summation would diverge. For example, the full zeta function exists at s = −1
1
+
2
+
3
+
…
,
{\textstyle 1+2+3+\ldots \,,}
partial sums would grow indefinitely large.
The zeta function values listed below include function values at the negative even numbers (s = −2−4 , etc.ζ (s ) = 0trivial zeros . The Riemann zeta function article includes a colour plot illustrating how the function varies over a continuous rectangular region of the complex plane. The successful characterisation of its non-trivial zeros in the wider plane is important in number theory, because of the Riemann hypothesis .
The Riemann zeta function at 0 and 1 [ ] At zero , one has
ζ
(
0
)
=
B
1
−
=
−
B
1
+
=
−
1
2
{\displaystyle \zeta (0)={B_{1}^{-}}=-{B_{1}^{+}}=-{\tfrac {1}{2}}\!}
At 1 there is a pole , so ζ (1) is not finite but the left and right limits are:
lim
ε
→
0
±
ζ
(
1
+
ε
)
=
±
∞
{\displaystyle \lim _{\varepsilon \to 0^{\pm }}\zeta (1+\varepsilon )=\pm \infty }
Since it is a pole of first order, it has a
complex residue
lim
ε
→
0
ε
ζ
(
1
+
ε
)
=
1
.
{\displaystyle \lim _{\varepsilon \to 0}\varepsilon \zeta (1+\varepsilon )=1\,.}
Positive integers [ ] Even positive integers [ ] For the even positive integers, one has the relationship to the Bernoulli numbers :
ζ
(
2
n
)
=
(
−
1
)
n
+
1
(
2
π
)
2
n
B
2
n
2
(
2
n
)
!
{\displaystyle \zeta (2n)=(-1)^{n+1}{\frac {(2\pi )^{2n}B_{2n}}{2(2n)!}}\!}
for
n
∈
N
{\displaystyle n\in \mathbb {N} }
The computation of ζ (2) is known as the Basel problem . The value of ζ (4) is related to the Stefan–Boltzmann law and Wien approximation in physics. The first few values are given by:
ζ
(
2
)
=
1
+
1
2
2
+
1
3
2
+
⋯
=
π
2
6
ζ
(
4
)
=
1
+
1
2
4
+
1
3
4
+
⋯
=
π
4
90
ζ
(
6
)
=
1
+
1
2
6
+
1
3
6
+
⋯
=
π
6
945
ζ
(
8
)
=
1
+
1
2
8
+
1
3
8
+
⋯
=
π
8
9450
ζ
(
10
)
=
1
+
1
2
10
+
1
3
10
+
⋯
=
π
10
93555
ζ
(
12
)
=
1
+
1
2
12
+
1
3
12
+
⋯
=
691
π
12
638512875
ζ
(
14
)
=
1
+
1
2
14
+
1
3
14
+
⋯
=
2
π
14
18243225
.
{\displaystyle {\begin{aligned}\zeta (2)&=1+{\frac {1}{2^{2}}}+{\frac {1}{3^{2}}}+\cdots ={\frac {\pi ^{2}}{6}}\\[4pt]\zeta (4)&=1+{\frac {1}{2^{4}}}+{\frac {1}{3^{4}}}+\cdots ={\frac {\pi ^{4}}{90}}\\[4pt]\zeta (6)&=1+{\frac {1}{2^{6}}}+{\frac {1}{3^{6}}}+\cdots ={\frac {\pi ^{6}}{945}}\\[4pt]\zeta (8)&=1+{\frac {1}{2^{8}}}+{\frac {1}{3^{8}}}+\cdots ={\frac {\pi ^{8}}{9450}}\\[4pt]\zeta (10)&=1+{\frac {1}{2^{10}}}+{\frac {1}{3^{10}}}+\cdots ={\frac {\pi ^{10}}{93555}}\\[4pt]\zeta (12)&=1+{\frac {1}{2^{12}}}+{\frac {1}{3^{12}}}+\cdots ={\frac {691\pi ^{12}}{638512875}}\\[4pt]\zeta (14)&=1+{\frac {1}{2^{14}}}+{\frac {1}{3^{14}}}+\cdots ={\frac {2\pi ^{14}}{18243225}}\,.\end{aligned}}}
Taking the limit
n
→
∞
{\displaystyle n\rightarrow \infty }
ζ
(
∞
)
=
1
{\displaystyle \zeta (\infty )=1}
Selected values for even integers
Value
Decimal expansion
Source
ζ
(
2
)
{\displaystyle \zeta (2)}
934 066 848 226 4364 OEIS : A013661
ζ
(
4
)
{\displaystyle \zeta (4)}
323 233 711 138 1915 OEIS : A013662
ζ
(
6
)
{\displaystyle \zeta (6)}
343 061 984 449 1397 OEIS : A013664
ζ
(
8
)
{\displaystyle \zeta (8)}
077 356 197 944 3393 OEIS : A013666
ζ
(
10
)
{\displaystyle \zeta (10)}
994 575 127 818 0853 OEIS : A013668
ζ
(
12
)
{\displaystyle \zeta (12)}
246 086 553 308 0482 OEIS : A013670
ζ
(
14
)
{\displaystyle \zeta (14)}
061 248 135 058 7048 OEIS : A013672
The relationship between zeta at the positive even integers and the Bernoulli numbers may be written as
A
n
ζ
(
2
n
)
=
π
2
n
B
n
{\displaystyle A_{n}\zeta (2n)=\pi ^{2n}B_{n}}
where
A
n
{\displaystyle A_{n}}
B
n
{\displaystyle B_{n}}
n
{\displaystyle n}
OEIS : A002432 OEIS : A046988 OEIS . Some of these values are reproduced below:
coefficients
n
A
B
1
6
1
2
90
1
3
945
1
4
9450
1
5
93555
1
6
638512875
691
7
18243225
2
8
325641566250
3617
9
38979295480125
43867
10
1531329465290625
174611
11
13447856940643125
155366
12
201919571963756521875
236364091
13
11094481976030578125
1315862
14
564653660170076273671875
6785560294
15
5660878804669082674070015625
6892673020804
16
62490220571022341207266406250
7709321041217
17
12130454581433748587292890625
151628697551
If we let
η
n
=
B
n
/
A
n
{\displaystyle \eta _{n}=B_{n}/A_{n}}
π
2
n
{\displaystyle \pi ^{2n}}
ζ
(
2
n
)
=
∑
ℓ
=
1
∞
1
ℓ
2
n
=
η
n
π
2
n
{\displaystyle \zeta (2n)=\sum _{\ell =1}^{\infty }{\frac {1}{\ell ^{2n}}}=\eta _{n}\pi ^{2n}}
then we find recursively,
η
1
=
1
/
6
η
n
=
∑
ℓ
=
1
n
−
1
(
−
1
)
ℓ
−
1
η
n
−
ℓ
(
2
ℓ
+
1
)
!
+
(
−
1
)
n
+
1
n
(
2
n
+
1
)
!
{\displaystyle {\begin{aligned}\eta _{1}&=1/6\\\eta _{n}&=\sum _{\ell =1}^{n-1}(-1)^{\ell -1}{\frac {\eta _{n-\ell }}{(2\ell +1)!}}+(-1)^{n+1}{\frac {n}{(2n+1)!}}\end{aligned}}}
This recurrence relation may be derived from that for the Bernoulli numbers .
Also, there is another recurrence:
ζ
(
2
n
)
=
1
n
+
1
2
∑
k
=
1
n
−
1
ζ
(
2
k
)
ζ
(
2
n
−
2
k
)
for
n
>
1
{\displaystyle \zeta (2n)={\frac {1}{n+{\frac {1}{2}}}}\sum _{k=1}^{n-1}\zeta (2k)\zeta (2n-2k)\quad {\text{ for }}\quad n>1}
which can be proved, using that
d
d
x
cot
(
x
)
=
−
1
−
cot
2
(
x
)
{\displaystyle {\frac {d}{dx}}\cot(x)=-1-\cot ^{2}(x)}
The values of the zeta function at non-negative even integers have the generating function :
∑
n
=
0
∞
ζ
(
2
n
)
x
2
n
=
−
π
x
2
cot
(
π
x
)
=
−
1
2
+
π
2
6
x
2
+
π
4
90
x
4
+
π
6
945
x
6
+
⋯
{\displaystyle \sum _{n=0}^{\infty }\zeta (2n)x^{2n}=-{\frac {\pi x}{2}}\cot(\pi x)=-{\frac {1}{2}}+{\frac {\pi ^{2}}{6}}x^{2}+{\frac {\pi ^{4}}{90}}x^{4}+{\frac {\pi ^{6}}{945}}x^{6}+\cdots }
Since
lim
n
→
∞
ζ
(
2
n
)
=
1
{\displaystyle \lim _{n\rightarrow \infty }\zeta (2n)=1}
The formula also shows that for
n
∈
N
,
n
→
∞
{\displaystyle n\in \mathbb {N} ,n\rightarrow \infty }
,
|
B
2
n
|
∼
(
2
n
)
!
2
(
2
π
)
2
n
{\displaystyle \left|B_{2n}\right|\sim {\frac {(2n)!\,2}{\;~(2\pi )^{2n}\,}}}
Odd positive integers [ ] The sum of the harmonic series is infinite.
ζ
(
1
)
=
1
+
1
2
+
1
3
+
⋯
=
∞
{\displaystyle \zeta (1)=1+{\frac {1}{2}}+{\frac {1}{3}}+\cdots =\infty \!}
The value ζ (3)Apéry's constant and has a role in the electron's gyromagnetic ratio.
The value ζ (3)Planck's law .
These and additional values are:
Selected values for odd integers
Value
Decimal expansion
Source
ζ
(
3
)
{\displaystyle \zeta (3)}
056 903 159 594 2853 OEIS : A02117
ζ
(
5
)
{\displaystyle \zeta (5)}
927 755 143 369 9263 OEIS : A013663
ζ
(
7
)
{\displaystyle \zeta (7)}
349 277 381 922 8268 OEIS : A013665
ζ
(
9
)
{\displaystyle \zeta (9)}
008 392 826 082 2144 OEIS : A013667
ζ
(
11
)
{\displaystyle \zeta (11)}
494 188 604 119 4645 OEIS : A013669
ζ
(
13
)
{\displaystyle \zeta (13)}
122 713 347 578 4891 OEIS : A013671
ζ
(
15
)
{\displaystyle \zeta (15)}
030 588 236 307 0204 OEIS : A013673
It is known that ζ (3)Apéry's theorem ) and that infinitely many of the numbers ζ (2n + 1) : n ∈
N
{\displaystyle \mathbb {N} }
[1] ζ (5), ζ (7), ζ (9), or ζ (11)[2]
The positive odd integers of the zeta function appear in physics, specifically correlation functions of antiferromagnetic XXX spin chain .[3]
Most of the identities following below are provided by Simon Plouffe . They are notable in that they converge quite rapidly, giving almost three digits of precision per iteration, and are thus useful for high-precision calculations.
ζ (5)[ ] Plouffe gives the following identities
ζ
(
5
)
=
1
294
π
5
−
72
35
∑
n
=
1
∞
1
n
5
(
e
2
π
n
−
1
)
−
2
35
∑
n
=
1
∞
1
n
5
(
e
2
π
n
+
1
)
ζ
(
5
)
=
12
∑
n
=
1
∞
1
n
5
sinh
(
π
n
)
−
39
20
∑
n
=
1
∞
1
n
5
(
e
2
π
n
−
1
)
−
1
20
∑
n
=
1
∞
1
n
5
(
e
2
π
n
+
1
)
{\displaystyle {\begin{aligned}\zeta (5)&={\frac {1}{294}}\pi ^{5}-{\frac {72}{35}}\sum _{n=1}^{\infty }{\frac {1}{n^{5}(e^{2\pi n}-1)}}-{\frac {2}{35}}\sum _{n=1}^{\infty }{\frac {1}{n^{5}(e^{2\pi n}+1)}}\\\zeta (5)&=12\sum _{n=1}^{\infty }{\frac {1}{n^{5}\sinh(\pi n)}}-{\frac {39}{20}}\sum _{n=1}^{\infty }{\frac {1}{n^{5}(e^{2\pi n}-1)}}-{\frac {1}{20}}\sum _{n=1}^{\infty }{\frac {1}{n^{5}(e^{2\pi n}+1)}}\end{aligned}}}
ζ (7)[ ]
ζ
(
7
)
=
19
56700
π
7
−
2
∑
n
=
1
∞
1
n
7
(
e
2
π
n
−
1
)
{\displaystyle \zeta (7)={\frac {19}{56700}}\pi ^{7}-2\sum _{n=1}^{\infty }{\frac {1}{n^{7}(e^{2\pi n}-1)}}\!}
Note that the sum is in the form of a Lambert series .
ζ (2n + 1)[ ] By defining the quantities
S
±
(
s
)
=
∑
n
=
1
∞
1
n
s
(
e
2
π
n
±
1
)
{\displaystyle S_{\pm }(s)=\sum _{n=1}^{\infty }{\frac {1}{n^{s}(e^{2\pi n}\pm 1)}}}
a series of relationships can be given in the form
0
=
A
n
ζ
(
n
)
−
B
n
π
n
+
C
n
S
−
(
n
)
+
D
n
S
+
(
n
)
{\displaystyle 0=A_{n}\zeta (n)-B_{n}\pi ^{n}+C_{n}S_{-}(n)+D_{n}S_{+}(n)}
where A n B n C n D n
coefficients
n
A
B
C
D
3
180
7
360
0
5
1470
5
3024
84
7
56700
19
113400
0
9
18523890
625
37122624
74844
11
425675250
1453
851350500
0
13
257432175
89
514926720
62370
15
390769879500
13687
781539759000
0
17
1904417007743250
6758333
3808863131673600
29116187100
19
21438612514068750
7708537
42877225028137500
0
21
1881063815762259253125
68529640373
3762129424572110592000
1793047592085750
These integer constants may be expressed as sums over Bernoulli numbers, as given in (Vepstas, 2006) below.
A fast algorithm for the calculation of Riemann's zeta function for any integer argument is given by E. A. Karatsuba.[4] [5] [6]
Negative integers [ ] In general, for negative integers (and also zero), one has
ζ
(
−
n
)
=
(
−
1
)
n
B
n
+
1
n
+
1
{\displaystyle \zeta (-n)=(-1)^{n}{\frac {B_{n+1}}{n+1}}}
The so-called "trivial zeros" occur at the negative even integers:
ζ
(
−
2
n
)
=
0
{\displaystyle \zeta (-2n)=0}
(
Ramanujan summation )
The first few values for negative odd integers are
ζ
(
−
1
)
=
−
1
12
ζ
(
−
3
)
=
1
120
ζ
(
−
5
)
=
−
1
252
ζ
(
−
7
)
=
1
240
ζ
(
−
9
)
=
−
1
132
ζ
(
−
11
)
=
691
32760
ζ
(
−
13
)
=
−
1
12
{\displaystyle {\begin{aligned}\zeta (-1)&=-{\frac {1}{12}}\\[4pt]\zeta (-3)&={\frac {1}{120}}\\[4pt]\zeta (-5)&=-{\frac {1}{252}}\\[4pt]\zeta (-7)&={\frac {1}{240}}\\[4pt]\zeta (-9)&=-{\frac {1}{132}}\\[4pt]\zeta (-11)&={\frac {691}{32760}}\\[4pt]\zeta (-13)&=-{\frac {1}{12}}\end{aligned}}}
However, just like the Bernoulli numbers , these do not stay small for increasingly negative odd values. For details on the first value, see 1 + 2 + 3 + 4 + · · · .
So ζ (m ) can be used as the definition of all (including those for index 0 and 1) Bernoulli numbers.
Derivatives [ ] The derivative of the zeta function at the negative even integers is given by
ζ
′
(
−
2
n
)
=
(
−
1
)
n
(
2
n
)
!
2
(
2
π
)
2
n
ζ
(
2
n
+
1
)
.
{\displaystyle \zeta ^{\prime }(-2n)=(-1)^{n}{\frac {(2n)!}{2(2\pi )^{2n}}}\zeta (2n+1)\,.}
The first few values of which are
ζ
′
(
−
2
)
=
−
ζ
(
3
)
4
π
2
ζ
′
(
−
4
)
=
3
4
π
4
ζ
(
5
)
ζ
′
(
−
6
)
=
−
45
8
π
6
ζ
(
7
)
ζ
′
(
−
8
)
=
315
4
π
8
ζ
(
9
)
.
{\displaystyle {\begin{aligned}\zeta ^{\prime }(-2)&=-{\frac {\zeta (3)}{4\pi ^{2}}}\\[4pt]\zeta ^{\prime }(-4)&={\frac {3}{4\pi ^{4}}}\zeta (5)\\[4pt]\zeta ^{\prime }(-6)&=-{\frac {45}{8\pi ^{6}}}\zeta (7)\\[4pt]\zeta ^{\prime }(-8)&={\frac {315}{4\pi ^{8}}}\zeta (9)\,.\end{aligned}}}
One also has
ζ
′
(
0
)
=
−
1
2
ln
(
2
π
)
ζ
′
(
−
1
)
=
1
12
−
ln
A
ζ
′
(
2
)
=
1
6
π
2
(
γ
+
ln
2
−
12
ln
A
+
ln
π
)
{\displaystyle {\begin{aligned}\zeta ^{\prime }(0)&=-{\frac {1}{2}}\ln(2\pi )\\[4pt]\zeta ^{\prime }(-1)&={\frac {1}{12}}-\ln A\\[4pt]\zeta ^{\prime }(2)&={\frac {1}{6}}\pi ^{2}(\gamma +\ln 2-12\ln A+\ln \pi )\end{aligned}}}
where A is the Glaisher–Kinkelin constant .
From the logarithmic derivative of the functional equation,
2
ζ
′
(
1
/
2
)
ζ
(
1
/
2
)
=
log
(
2
π
)
+
π
cos
(
π
/
4
)
2
sin
(
π
/
4
)
−
Γ
′
(
1
/
2
)
Γ
(
1
/
2
)
=
log
(
2
π
)
+
π
2
+
2
log
2
+
γ
.
{\displaystyle 2{\frac {\zeta '(1/2)}{\zeta (1/2)}}=\log(2\pi )+{\frac {\pi \cos(\pi /4)}{2\sin(\pi /4)}}-{\frac {\Gamma '(1/2)}{\Gamma (1/2)}}=\log(2\pi )+{\frac {\pi }{2}}+2\log 2+\gamma \,.}
Selected derivatives
Value
Decimal expansion
Source
ζ
′
(
3
)
{\displaystyle \zeta '(3)}
126 242 885 636 853 33 OEIS : A244115
ζ
′
(
2
)
{\displaystyle \zeta '(2)}
548 254 315 843 753 70 OEIS : A073002
ζ
′
(
0
)
{\displaystyle \zeta '(0)}
938 533 204 672 741 78 OEIS : A075700
ζ
′
(
−
1
2
)
{\displaystyle \zeta '(-{\tfrac {1}{2}})}
854 339 599 947 607 34 OEIS : A271854
ζ
′
(
−
1
)
{\displaystyle \zeta '(-1)}
421 143 700 450 929 21 OEIS : A084448
ζ
′
(
−
2
)
{\displaystyle \zeta '(-2)}
448 457 058 393 270 780 OEIS : A240966
ζ
′
(
−
3
)
{\displaystyle \zeta '(-3)}
378 576 357 774 301 1444 OEIS : A259068
ζ
′
(
−
4
)
{\displaystyle \zeta '(-4)}
983 811 450 268 624 2806 OEIS : A259069
ζ
′
(
−
5
)
{\displaystyle \zeta '(-5)}
572 985 980 198 635 204 99 OEIS : A259070
ζ
′
(
−
6
)
{\displaystyle \zeta '(-6)}
899 759 143 515 937 4506 OEIS : A259071
ζ
′
(
−
7
)
{\displaystyle \zeta '(-7)}
728 642 680 159 240 652 46 OEIS : A259072
ζ
′
(
−
8
)
{\displaystyle \zeta '(-8)}
316 161 985 602 247 3595 OEIS : A259073
Series involving ζ (n ) [ ] The following sums can be derived from the generating function:
∑
k
=
2
∞
ζ
(
k
)
x
k
−
1
=
−
ψ
0
(
1
−
x
)
−
γ
{\displaystyle \sum _{k=2}^{\infty }\zeta (k)x^{k-1}=-\psi _{0}(1-x)-\gamma }
where
ψ 0 is the
digamma function .
∑
k
=
2
∞
(
ζ
(
k
)
−
1
)
=
1
∑
k
=
1
∞
(
ζ
(
2
k
)
−
1
)
=
3
4
∑
k
=
1
∞
(
ζ
(
2
k
+
1
)
−
1
)
=
1
4
∑
k
=
2
∞
(
−
1
)
k
(
ζ
(
k
)
−
1
)
=
1
2
{\displaystyle {\begin{aligned}\sum _{k=2}^{\infty }(\zeta (k)-1)&=1\\[4pt]\sum _{k=1}^{\infty }(\zeta (2k)-1)&={\frac {3}{4}}\\[4pt]\sum _{k=1}^{\infty }(\zeta (2k+1)-1)&={\frac {1}{4}}\\[4pt]\sum _{k=2}^{\infty }(-1)^{k}(\zeta (k)-1)&={\frac {1}{2}}\end{aligned}}}
Series related to the Euler–Mascheroni constant (denoted by γ
∑
k
=
2
∞
(
−
1
)
k
ζ
(
k
)
k
=
γ
∑
k
=
2
∞
ζ
(
k
)
−
1
k
=
1
−
γ
∑
k
=
2
∞
(
−
1
)
k
ζ
(
k
)
−
1
k
=
ln
2
+
γ
−
1
{\displaystyle {\begin{aligned}\sum _{k=2}^{\infty }(-1)^{k}{\frac {\zeta (k)}{k}}&=\gamma \\[4pt]\sum _{k=2}^{\infty }{\frac {\zeta (k)-1}{k}}&=1-\gamma \\[4pt]\sum _{k=2}^{\infty }(-1)^{k}{\frac {\zeta (k)-1}{k}}&=\ln 2+\gamma -1\end{aligned}}}
and using the principal value
ζ
(
k
)
=
lim
ε
→
0
ζ
(
k
+
ε
)
+
ζ
(
k
−
ε
)
2
{\displaystyle \zeta (k)=\lim _{\varepsilon \to 0}{\frac {\zeta (k+\varepsilon )+\zeta (k-\varepsilon )}{2}}}
which of course affects only the value at 1, these formulae can be stated as
∑
k
=
1
∞
(
−
1
)
k
ζ
(
k
)
k
=
0
∑
k
=
1
∞
ζ
(
k
)
−
1
k
=
0
∑
k
=
1
∞
(
−
1
)
k
ζ
(
k
)
−
1
k
=
ln
2
{\displaystyle {\begin{aligned}\sum _{k=1}^{\infty }(-1)^{k}{\frac {\zeta (k)}{k}}&=0\\[4pt]\sum _{k=1}^{\infty }{\frac {\zeta (k)-1}{k}}&=0\\[4pt]\sum _{k=1}^{\infty }(-1)^{k}{\frac {\zeta (k)-1}{k}}&=\ln 2\end{aligned}}}
and show that they depend on the principal value of ζ (1) = γ
Nontrivial zeros [ ] Zeros of the Riemann zeta except negative even integers are called "nontrivial zeros". See Andrew Odlyzko 's website for their tables and bibliographies.
Ratios [ ] Although evaluating particular values of the zeta function is difficult, often certain ratios can be found by inserting particular values of the gamma function into the functional equation
ζ
(
s
)
=
2
s
π
s
−
1
sin
(
π
s
2
)
Γ
(
1
−
s
)
ζ
(
1
−
s
)
{\displaystyle \zeta (s)=2^{s}\pi ^{s-1}\sin \left({\frac {\pi s}{2}}\right)\Gamma (1-s)\zeta (1-s)}
We have simple relations for half-integer arguments
ζ
(
3
/
2
)
ζ
(
−
1
/
2
)
=
−
4
π
ζ
(
5
/
2
)
ζ
(
−
3
/
2
)
=
−
16
π
2
3
ζ
(
7
/
2
)
ζ
(
−
5
/
2
)
=
64
π
3
15
ζ
(
9
/
2
)
ζ
(
−
7
/
2
)
=
256
π
4
105
{\displaystyle {\begin{aligned}{\frac {\zeta (3/2)}{\zeta (-1/2)}}&=-4\pi \\{\frac {\zeta (5/2)}{\zeta (-3/2)}}&=-{\frac {16\pi ^{2}}{3}}\\{\frac {\zeta (7/2)}{\zeta (-5/2)}}&={\frac {64\pi ^{3}}{15}}\\{\frac {\zeta (9/2)}{\zeta (-7/2)}}&={\frac {256\pi ^{4}}{105}}\end{aligned}}}
Other examples follow for more complicated evaluations and relations of the gamma function. For example a consequence of the relation
Γ
(
3
4
)
=
(
π
2
)
1
4
AGM
(
2
,
1
)
1
2
{\displaystyle \Gamma \left({\tfrac {3}{4}}\right)=\left({\tfrac {\pi }{2}}\right)^{\tfrac {1}{4}}{\operatorname {AGM} \left({\sqrt {2}},1\right)}^{\tfrac {1}{2}}}
is the zeta ratio relation
ζ
(
3
/
4
)
ζ
(
1
/
4
)
=
2
π
(
2
−
2
)
AGM
(
2
,
1
)
{\displaystyle {\frac {\zeta (3/4)}{\zeta (1/4)}}=2{\sqrt {\frac {\pi }{(2-{\sqrt {2}})\operatorname {AGM} \left({\sqrt {2}},1\right)}}}}
where AGM is the arithmetic–geometric mean . In a similar vein, it is possible to form radical relations, such as from
Γ
(
1
5
)
2
Γ
(
1
10
)
Γ
(
3
10
)
=
1
+
5
2
7
10
5
4
{\displaystyle {\frac {\Gamma \left({\frac {1}{5}}\right)^{2}}{\Gamma \left({\frac {1}{10}}\right)\Gamma \left({\frac {3}{10}}\right)}}={\frac {\sqrt {1+{\sqrt {5}}}}{2^{\tfrac {7}{10}}{\sqrt[{4}]{5}}}}}
the analogous zeta relation is
ζ
(
1
/
5
)
2
ζ
(
7
/
10
)
ζ
(
9
/
10
)
ζ
(
1
/
10
)
ζ
(
3
/
10
)
ζ
(
4
/
5
)
2
=
(
5
−
5
)
(
10
+
5
+
5
)
10
⋅
2
3
10
{\displaystyle {\frac {\zeta (1/5)^{2}\zeta (7/10)\zeta (9/10)}{\zeta (1/10)\zeta (3/10)\zeta (4/5)^{2}}}={\frac {(5-{\sqrt {5}})\left({\sqrt {10}}+{\sqrt {5+{\sqrt {5}}}}\right)}{10\cdot 2^{\tfrac {3}{10}}}}}
References [ ]
^ Rivoal, T. (2000). "La fonction zeta de Riemann prend une infinité de valeurs irrationnelles aux entiers impairs". Comptes Rendus de l'Académie des Sciences, Série I . 331 : 267–270. arXiv :math/0008051 Bibcode :2000CRASM.331..267R . doi :10.1016/S0764-4442(00)01624-4 . S2CID 119678120 . ^ W. Zudilin (2001). "One of the numbers ζ (5), ζ (7), ζ (9), ζ (11) is irrational". Russ. Math. Surv . 56 (4): 774–776. Bibcode :2001RuMaS..56..774Z . doi :10.1070/rm2001v056n04abeh000427 . ^ Boos, H.E.; Korepin, V.E.; Nishiyama, Y.; Shiroishi, M. (2002). "Quantum correlations and number theory". J. Phys. A . 35 (20): 4443–4452. arXiv :cond-mat/0202346 Bibcode :2002JPhA...35.4443B . doi :10.1088/0305-4470/35/20/305 . S2CID 119143600 . ^ Karatsuba, E. A. (1995). "Fast calculation of the Riemann zeta function ζ (s ) for integer values of the argument s " . Probl. Perdachi Inf . 31 (4): 69–80. MR 1367927 . ^ E. A. Karatsuba: Fast computation of the Riemann zeta function for integer argument. Dokl. Math. Vol.54, No.1, p. 626 (1996).
^ E. A. Karatsuba: Fast evaluation of ζ (3). Probl. Inf. Transm. Vol.29, No.1, pp. 58–62 (1993).
Further reading [ ] Ciaurri, Óscar; Navas, Luis M.; Ruiz, Francisco J.; Varona, Juan L. (May 2015). "A Simple Computation of ζ (2k )". The American Mathematical Monthly . 122 (5): 444–451. doi :10.4169/amer.math.monthly.122.5.444 . JSTOR 10.4169/amer.math.monthly.122.5.444 . S2CID 207521195 . Simon Plouffe , "Identities inspired from Ramanujan Notebooks ", (1998).Simon Plouffe , "Identities inspired by Ramanujan Notebooks part 2 PDF " (2006).Vepstas, Linas (2006). "On Plouffe's Ramanujan identities" (PDF) . The Ramanujan Journal . 27 (3): 387–408. arXiv :math.NT/0609775 doi :10.1007/s11139-011-9335-9 . S2CID 8789411 . Zudilin, Wadim (2001). "One of the Numbers ζ (5), ζ (7), ζ (9), ζ (11) Is Irrational". Russian Mathematical Surveys 56 (4): 774–776. Bibcode :2001RuMaS..56..774Z . doi :10.1070/RM2001v056n04ABEH000427 . MR 1861452 .PDF PDF Russian PS Russian Nontrival zeros reference by Andrew Odlyzko :
![{\displaystyle {\begin{aligned}\zeta (2)&=1+{\frac {1}{2^{2}}}+{\frac {1}{3^{2}}}+\cdots ={\frac {\pi ^{2}}{6}}\\[4pt]\zeta (4)&=1+{\frac {1}{2^{4}}}+{\frac {1}{3^{4}}}+\cdots ={\frac {\pi ^{4}}{90}}\\[4pt]\zeta (6)&=1+{\frac {1}{2^{6}}}+{\frac {1}{3^{6}}}+\cdots ={\frac {\pi ^{6}}{945}}\\[4pt]\zeta (8)&=1+{\frac {1}{2^{8}}}+{\frac {1}{3^{8}}}+\cdots ={\frac {\pi ^{8}}{9450}}\\[4pt]\zeta (10)&=1+{\frac {1}{2^{10}}}+{\frac {1}{3^{10}}}+\cdots ={\frac {\pi ^{10}}{93555}}\\[4pt]\zeta (12)&=1+{\frac {1}{2^{12}}}+{\frac {1}{3^{12}}}+\cdots ={\frac {691\pi ^{12}}{638512875}}\\[4pt]\zeta (14)&=1+{\frac {1}{2^{14}}}+{\frac {1}{3^{14}}}+\cdots ={\frac {2\pi ^{14}}{18243225}}\,.\end{aligned}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/3596c7957d1412be9c691a316b4fb31cab6703fb)
![{\displaystyle {\begin{aligned}\zeta (-1)&=-{\frac {1}{12}}\\[4pt]\zeta (-3)&={\frac {1}{120}}\\[4pt]\zeta (-5)&=-{\frac {1}{252}}\\[4pt]\zeta (-7)&={\frac {1}{240}}\\[4pt]\zeta (-9)&=-{\frac {1}{132}}\\[4pt]\zeta (-11)&={\frac {691}{32760}}\\[4pt]\zeta (-13)&=-{\frac {1}{12}}\end{aligned}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/1f4a2858bf5436ce8cadf3a5906f64b006d648de)
![{\displaystyle {\begin{aligned}\zeta ^{\prime }(-2)&=-{\frac {\zeta (3)}{4\pi ^{2}}}\\[4pt]\zeta ^{\prime }(-4)&={\frac {3}{4\pi ^{4}}}\zeta (5)\\[4pt]\zeta ^{\prime }(-6)&=-{\frac {45}{8\pi ^{6}}}\zeta (7)\\[4pt]\zeta ^{\prime }(-8)&={\frac {315}{4\pi ^{8}}}\zeta (9)\,.\end{aligned}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/4734f2991ac4b8d15765ec1637a6a93df4841992)
![{\displaystyle {\begin{aligned}\zeta ^{\prime }(0)&=-{\frac {1}{2}}\ln(2\pi )\\[4pt]\zeta ^{\prime }(-1)&={\frac {1}{12}}-\ln A\\[4pt]\zeta ^{\prime }(2)&={\frac {1}{6}}\pi ^{2}(\gamma +\ln 2-12\ln A+\ln \pi )\end{aligned}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/c83b15faa8cfb04ca281721bb4a3069c6a41da27)
![{\displaystyle {\begin{aligned}\sum _{k=2}^{\infty }(\zeta (k)-1)&=1\\[4pt]\sum _{k=1}^{\infty }(\zeta (2k)-1)&={\frac {3}{4}}\\[4pt]\sum _{k=1}^{\infty }(\zeta (2k+1)-1)&={\frac {1}{4}}\\[4pt]\sum _{k=2}^{\infty }(-1)^{k}(\zeta (k)-1)&={\frac {1}{2}}\end{aligned}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/64c161575e1ebb9fcf86ff81945618c698e7c8a0)
![{\displaystyle {\begin{aligned}\sum _{k=2}^{\infty }(-1)^{k}{\frac {\zeta (k)}{k}}&=\gamma \\[4pt]\sum _{k=2}^{\infty }{\frac {\zeta (k)-1}{k}}&=1-\gamma \\[4pt]\sum _{k=2}^{\infty }(-1)^{k}{\frac {\zeta (k)-1}{k}}&=\ln 2+\gamma -1\end{aligned}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/a8fce86fde721036a8ddf8cd1061cbcbe8b164bb)
![{\displaystyle {\begin{aligned}\sum _{k=1}^{\infty }(-1)^{k}{\frac {\zeta (k)}{k}}&=0\\[4pt]\sum _{k=1}^{\infty }{\frac {\zeta (k)-1}{k}}&=0\\[4pt]\sum _{k=1}^{\infty }(-1)^{k}{\frac {\zeta (k)-1}{k}}&=\ln 2\end{aligned}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/a812f5dc324ba048e1a08c9cd90253ada721735f)
![{\displaystyle {\frac {\Gamma \left({\frac {1}{5}}\right)^{2}}{\Gamma \left({\frac {1}{10}}\right)\Gamma \left({\frac {3}{10}}\right)}}={\frac {\sqrt {1+{\sqrt {5}}}}{2^{\tfrac {7}{10}}{\sqrt[{4}]{5}}}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/13c849db6b4d10213e21b6f070927b56d2e7148e)
