Particular values of the Riemann zeta function


This article gives some specific values of the Riemann zeta function, including values at integer arguments, and some series involving them.

The Riemann zeta function at 0 and 1

At zero, one has
At 1 there is a pole, so ζ is not finite but the left and right limits are:
Since it is a pole of first order, its principal value exists and is equal to the Euler–Mascheroni constant γ = 0.57721 56649+.

Positive integers

Even positive integers

For the even positive integers, one has the relationship to the Bernoulli numbers:
for. The first few values are given by:
Taking the limit, one obtains.
The relationship between zeta at the positive even integers and the Bernoulli numbers may be written as
where and are integers for all even. These are given by the integer sequences and, respectively, in OEIS. Some of these values are reproduced below:
nAB
161
2901
39451
494501
5935551
6638512875691
7182432252
83256415662503617
93897929548012543867
101531329465290625174611
1113447856940643125155366
12201919571963756521875236364091
13110944819760305781251315862
145646536601700762736718756785560294
1556608788046690826740700156256892673020804
16624902205710223412072664062507709321041217
1712130454581433748587292890625151628697551

If we let be the coefficient of as above,
then we find recursively,
This recurrence relation may be derived from that for the Bernoulli numbers.
Also, there is another recurrence:
which can be proved, using that
The values of the zeta function at non-negative even integers have the generating function:
Since
The formula also shows that for ,

Odd positive integers

For the first few odd natural numbers one has
It is known that is irrational and that infinitely many of the numbers, are irrational. 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.
The positive odd integers of the zeta function appear in physics, specifically correlation functions of antiferromagnetic XXX spin chain.
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''(7)

Note that the sum is in the form of a Lambert series.

''ζ''(2''n'' + 1)

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:
nABCD
318073600
514705302484
756700191134000
9185238906253712262474844
1142567525014538513505000
132574321758951492672062370
15390769879500136877815397590000
1719044170077432506758333380886313167360029116187100
19214386125140687507708537428772250281375000
2118810638157622592531256852964037337621294245721105920001793047592085750

These integer constants may be expressed as sums over Bernoulli numbers, as given in below.
A fast algorithm for the calculation of Riemann's zeta function for any integer argument is given by E. A. Karatsuba.

Negative integers

In general, for negative integers, 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 can be used as the definition of all Bernoulli numbers.

Derivatives

The derivative of the zeta function at the negative even integers is given by
The first few values of which are
One also has
and
where A is the Glaisher–Kinkelin constant.

Series involving ''ζ''(''n'')

The following sums can be derived from the generating function:
where is the digamma function.
Series related to the Euler–Mascheroni constant 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

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.