In mathematics, the Riemann zeta function is a function in complex analysis, which is also important in number theory. It is often denoted
\zeta(s)
s
\zeta(s)
The same equation in
s
s
s=1
s
s=-1
The zeta function values listed below include function values at the negative even numbers, for which 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.
At zero, one has
At 1 there is a pole, so ζ(1) is not finite but the left and right limits are:Since it is a pole of first order, it has a complex residue
For the even positive integers
n
Bn
The computation of
\zeta(2)
\zeta(4)
Taking the limit
n → infty
\zeta(infty)=1
Value | Decimal expansion | Source | |
---|---|---|---|
\zeta(2) | ... | ||
\zeta(4) | ... | ||
\zeta(6) | ... | ||
\zeta(8) | ... | ||
\zeta(10) | ... | ||
\zeta(12) | ... | ||
\zeta(14) | ... | ||
\zeta(16) | ... |
The relationship between zeta at the positive even integers and the Bernoulli numbers may be written as
where
An
Bn
n
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
ηn=Bn/An
\pi2n
This recurrence relation may be derived from that for the Bernoulli numbers.
Also, there is another recurrence:
which can be proved, using that
d | |
dx |
\cot(x)=-1-\cot2(x)
The values of the zeta function at non-negative even integers have the generating function:Since The formula also shows that for
n\inN,n → infty
The sum of the harmonic series is infinite.
The value is also known as Apéry's constant and has a role in the electron's gyromagnetic ratio.The value also appears in Planck's law.These and additional values are:
Value | Decimal expansion | Source | |
---|---|---|---|
\zeta(3) | ... | ||
\zeta(5) | ... | ||
\zeta(7) | ... | ||
\zeta(9) | ... | ||
\zeta(11) | ... | ||
\zeta(13) | ... | ||
\zeta(15) | ... |
It is known that is irrational (Apéry's theorem) and that infinitely many of the numbers, 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 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]
Note that the sum is in the form of a Lambert series.
By defining the quantities
a series of relationships can be given in the form
where An, Bn, Cn and Dn are positive integers. Plouffe gives a table of values:
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]
In general, for negative integers (and also zero), one has
The so-called "trivial zeros" occur at the negative even integers:
The first few values for negative odd integers are
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.
The derivative of the zeta function at the negative even integers is given by
The first few values of which are
One also has
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/\sqrt{2\pi}
infty!=\sqrt{2\pi}
From the logarithmic derivative of the functional equation,
Value | Decimal expansion | Source | |
---|---|---|---|
\zeta'(3) | ... | ||
\zeta'(2) | ... | ||
\zeta'(0) | ... | ||
\zeta'(-\tfrac{1}{2}) | ... | ||
\zeta'(-1) | ... | ||
\zeta'(-2) | ... | ||
\zeta'(-3) | ... | ||
\zeta'(-4) | ... | ||
\zeta'(-5) | ... | ||
\zeta'(-6) | ... | ||
\zeta'(-7) | ... | ||
\zeta'(-8) | ... |
The following sums can be derived from the generating function:where is the digamma function.
Series related to the Euler–Mascheroni constant (denoted by) are
and using the principal value which of course affects only the value at 1, these formulae can be stated as
and show that they depend on the principal value of
See main article: Riemann hypothesis.
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 . In other words, all known nontrivial zeros of the Riemann zeta are of the form where y is a real number. The following table contains the decimal expansion of Im(z) for the first few nontrivial zeros:
Decimal expansion of Im(z) | Source | |
---|---|---|
... | ||
... | ||
... | ||
... | ||
... | ||
... | ||
... | ||
... | ||
... | ||
... |
Andrew Odlyzko computed the first 2 million nontrivial zeros accurate to within 4, 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]
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
\zeta(s)=2s\pis-1\sin\left(
\pis | |
2 |
\right)\Gamma(1-s)\zeta(1-s)
We have simple relations for half-integer arguments
\begin{align} | \zeta(3/2) |
\zeta(-1/2) |
&=-4\pi\\
\zeta(5/2) | |
\zeta(-3/2) |
&=-
16\pi2 | \\ | |
3 |
\zeta(7/2) | |
\zeta(-5/2) |
&=
64\pi3 | \\ | |
15 |
\zeta(9/2) | |
\zeta(-7/2) |
&=
256\pi4 | |
105 |
\end{align}
Other examples follow for more complicated evaluations and relations of the gamma function. For example a consequence of the relation
\Gamma\left(\tfrac{3}{4}\right)=\left(\tfrac{\pi}{2}\right)\tfrac{1{4}}{\operatorname{AGM}\left(\sqrt2,1\right)}\tfrac{1{2}}
is the zeta ratio relation
\zeta(3/4) | |
\zeta(1/4) |
=2\sqrt{
\pi | |
(2-\sqrt{2 |
)\operatorname{AGM}\left(\sqrt2,1\right)}}
where AGM is the arithmetic–geometric mean. In a similar vein, it is possible to form radical relations, such as from
| |||||||
|
=
\sqrt{1+\sqrt{5 | |
the analogous zeta relation is
\zeta(1/5)2\zeta(7/10)\zeta(9/10) | |
\zeta(1/10)\zeta(3/10)\zeta(4/5)2 |
=
(5-\sqrt{5 | |
)\left(\sqrt{10}+\sqrt{5+\sqrt{5}}\right)}{10 ⋅ 2 |
\tfrac{3{10}}}