is a real number greater than one, the zeta function satisfies the equation
Explicit or numerically efficient formulae exist for
at integer arguments, all of which have real values, including this example.
It also includes derivatives and some series composed of the zeta function at integer arguments.
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
whose partial sums would grow indefinitely large.
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.
is related to the Stefan–Boltzmann law and Wien approximation in physics.
The relationship between zeta at the positive even integers and powers of pi may be written as
This recurrence relation may be derived from that for the Bernoulli numbers.
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.
[2] The positive odd integers of the zeta function appear in physics, specifically correlation functions of antiferromagnetic XXX spin chain.
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.
Plouffe gives a table of values: 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.
The so-called "trivial zeros" occur at the negative even integers:
(Ramanujan summation) 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.
So ζ(m) can be used as the definition of all (including those for index 0 and 1) Bernoulli numbers.
The first of these identities implies that the regularized product of the reciprocals of the positive integers is
Series related to the Euler–Mascheroni constant (denoted by γ) are
and show that they depend on the principal value of ζ(1) = γ .
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 + yi where y is a real number.
[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.
Other examples follow for more complicated evaluations and relations of the gamma function.