Zeta constants
This article gives some specific values of the Riemann zeta function, including values at integer arguments, and some series involving them.
At zero, one has
At 1 there is a pole, so ζ(1) is not defined 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+.
For the even positive integers, one has the relationship to the Bernoulli numbers:
for n ∈ N. The first few values are given by:
The relationship between zeta at the positive even integers and the Bernoulli numbers may be written as
where An and Bn are integers for all even n. These are given by the integer sequences 
and , respectively, in OEIS. Some of these values are reproduced below:
If we let ηn be the coefficient B/A 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
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