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
)
{\displaystyle \zeta (s)}
and is named after the mathematician
Bernhard Riemann . When the argument
s
{\displaystyle 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
)
{\displaystyle \zeta (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
{\displaystyle s}
above also holds when
s
{\displaystyle 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
=
1
{\displaystyle s=1}
. The
complex 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
{\displaystyle s}
, for which the corresponding summation would diverge. For example, the full zeta function exists at
s
=
−
1
{\displaystyle s=-1}
(and is therefore finite there), but the corresponding series would be
1
+
2
+
3
+
…
,
{\textstyle 1+2+3+\ldots \,,}
whose
partial sums would grow indefinitely large.
The zeta function values listed below include function values at the negative even numbers (s = −2 , −4 , etc. ), for which ζ (s ) = 0 and which make up the so-called trivial 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
n
{\displaystyle n}
, one has the relationship to the
Bernoulli numbers
B
n
{\displaystyle B_{n}}
:
ζ
(
n
)
=
(
−
1
)
n
2
+
1
(
2
π
)
n
B
n
2
(
n
!
)
.
{\displaystyle \zeta (n)=(-1)^{{\tfrac {n}{2}}+1}{\frac {(2\pi )^{n}B_{n}}{2(n!)}}\,.}
The computation of
ζ
(
2
)
{\displaystyle \zeta (2)}
is known as the
Basel problem . The value of
ζ
(
4
)
{\displaystyle \zeta (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
ζ
(
16
)
=
1
+
1
2
16
+
1
3
16
+
⋯
=
3617
π
16
325641566250
.
{\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}}\\[4pt]\zeta (16)&=1+{\frac {1}{2^{16}}}+{\frac {1}{3^{16}}}+\cdots ={\frac {3617\pi ^{16}}{325641566250}}\,.\end{aligned}}}
Taking the limit
n
→
∞
{\displaystyle n\rightarrow \infty }
, one obtains
ζ
(
∞
)
=
1
{\displaystyle \zeta (\infty )=1}
.
Selected values for even integers
Value
Decimal expansion
Source
ζ
(
2
)
{\displaystyle \zeta (2)}
1.644934 066 848 226 4364 ...
OEIS :
A013661
ζ
(
4
)
{\displaystyle \zeta (4)}
1.082323 233 711 138 1915 ...
OEIS :
A013662
ζ
(
6
)
{\displaystyle \zeta (6)}
1.017343 061 984 449 1397 ...
OEIS :
A013664
ζ
(
8
)
{\displaystyle \zeta (8)}
1.004077 356 197 944 3393 ...
OEIS :
A013666
ζ
(
10
)
{\displaystyle \zeta (10)}
1.000994 575 127 818 0853 ...
OEIS :
A013668
ζ
(
12
)
{\displaystyle \zeta (12)}
1.000246 086 553 308 0482 ...
OEIS :
A013670
ζ
(
14
)
{\displaystyle \zeta (14)}
1.000061 248 135 058 7048 ...
OEIS :
A013672
ζ
(
16
)
{\displaystyle \zeta (16)}
1.000015 282 259 408 6518 ...
OEIS :
A013674
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}}
and
B
n
{\displaystyle B_{n}}
are integers for all even
n
{\displaystyle n}
. These are given by the integer sequences
OEIS :
A002432 and
OEIS :
A046988 , respectively, in
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}}
be the coefficient of
π
2
n
{\displaystyle \pi ^{2n}}
as above,
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) is also known as
Apéry's constant and has a role in the electron's gyromagnetic ratio.
The value ζ (3) also appears in
Planck's law .
These and additional values are:
Selected values for odd integers
Value
Decimal expansion
Source
ζ
(
3
)
{\displaystyle \zeta (3)}
1.202056 903 159 594 2853 ...
OEIS :
A02117
ζ
(
5
)
{\displaystyle \zeta (5)}
1.036927 755 143 369 9263 ...
OEIS :
A013663
ζ
(
7
)
{\displaystyle \zeta (7)}
1.008349 277 381 922 8268 ...
OEIS :
A013665
ζ
(
9
)
{\displaystyle \zeta (9)}
1.002008 392 826 082 2144 ...
OEIS :
A013667
ζ
(
11
)
{\displaystyle \zeta (11)}
1.000494 188 604 119 4645 ...
OEIS :
A013669
ζ
(
13
)
{\displaystyle \zeta (13)}
1.000122 713 347 578 4891 ...
OEIS :
A013671
ζ
(
15
)
{\displaystyle \zeta (15)}
1.000030 588 236 307 0204 ...
OEIS :
A013673
It is known that ζ (3) is irrational (
Apéry's theorem ) and that infinitely many of the numbers ζ (2n + 1) : n ∈
N
{\displaystyle \mathbb {N} }
, are irrational.
[1] There are also results on the irrationality of values of the Riemann zeta function at the elements of certain subsets of the positive odd integers; for example, at least one of ζ (5), ζ (7), ζ (9), or ζ (11) is irrational.
[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.
Plouffe stated the following identities without proof.
[4] Proofs were later given by other authors.
[5]
ζ (5)
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 and D n are positive integers. Plouffe gives a table of values:
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.
[6]
[7]
[8]
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 . The first of these identities implies that the regularized product of the reciprocals of the positive integers is
1
/
2
π
{\displaystyle 1/{\sqrt {2\pi }}}
, thus the amusing "equation"
∞
!
=
2
π
{\displaystyle \infty !={\sqrt {2\pi }}}
.
[9]
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)}
−0.198126 242 885 636 853 33 ...
OEIS :
A244115
ζ
′
(
2
)
{\displaystyle \zeta '(2)}
−0.937548 254 315 843 753 70 ...
OEIS :
A073002
ζ
′
(
0
)
{\displaystyle \zeta '(0)}
−0.918938 533 204 672 741 78 ...
OEIS :
A075700
ζ
′
(
−
1
2
)
{\displaystyle \zeta '(-{\tfrac {1}{2}})}
−0.360854 339 599 947 607 34 ...
OEIS :
A271854
ζ
′
(
−
1
)
{\displaystyle \zeta '(-1)}
−0.165421 143 700 450 929 21 ...
OEIS :
A084448
ζ
′
(
−
2
)
{\displaystyle \zeta '(-2)}
−0.030448 457 058 393 270 780 ...
OEIS :
A240966
ζ
′
(
−
3
)
{\displaystyle \zeta '(-3)}
+0.005378 576 357 774 301 1444 ...
OEIS :
A259068
ζ
′
(
−
4
)
{\displaystyle \zeta '(-4)}
+0.007983 811 450 268 624 2806 ...
OEIS :
A259069
ζ
′
(
−
5
)
{\displaystyle \zeta '(-5)}
−0.000572 985 980 198 635 204 99 ...
OEIS :
A259070
ζ
′
(
−
6
)
{\displaystyle \zeta '(-6)}
−0.005899 759 143 515 937 4506 ...
OEIS :
A259071
ζ
′
(
−
7
)
{\displaystyle \zeta '(-7)}
−0.000728 642 680 159 240 652 46 ...
OEIS :
A259072
ζ
′
(
−
8
)
{\displaystyle \zeta '(-8)}
+0.008316 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 ) are
∑
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". The
Riemann hypothesis states that the real part of every nontrivial zero must be 1 / 2 . In other words, all known nontrivial zeros of the Riemann zeta are of the form z = 1 / 2 + y i where y is a real number. The following table contains the decimal expansion of Im(z ) for the first few nontrivial zeros:
Andrew Odlyzko computed the first 2 million nontrivial zeros accurate to within 4× 10 −9 , and the first 100 zeros accurate within 1000 decimal places. See their website for the tables and bibliographies.
[10]
[11]
A table of about 103 billion zeros with high precision (of ±2-102 ≈±2·10-31 ) is available for interactive access and download (although in a very inconvenient compressed format) via LMFDB .
[12]
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 (4): 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 .
S2CID
250734661 .
^ 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 . .
^
"Identities for Zeta(2*n+1)" .
^
"Formulas for Odd Zeta Values and Powers of Pi" .
^ 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).
^ Muñoz García, E.; Pérez Marco, R. (2008), "The Product Over All Primes is
4
π
2
{\displaystyle 4\pi ^{2}}
", Commun. Math. Phys. (277): 69–81 .
^ Odlyzko, Andrew.
"Tables of zeros of the Riemann zeta function" . Retrieved 7 September 2022 .
^ Odlyzko, Andrew.
"Papers on Zeros of the Riemann Zeta Function and Related Topics" . Retrieved 7 September 2022 .
^
LMFDB: Zeros of ζ(s )
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
Archived 2009-01-30 at the
Wayback Machine ", (1998).
Simon Plouffe , "
Identities inspired by Ramanujan Notebooks part 2
PDF
Archived 2011-09-26 at the
Wayback Machine " (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 .
S2CID
250734661 .
PDF
PDF Russian
PS Russian
Nontrival zeros reference by
Andrew Odlyzko :