# Riemann zeta function

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## Riemann zeta function

[′rē‚män ′zād·ə ‚fəŋk·shən]
(mathematics)
The complex function ζ(z) defined by an infinite series with n th term e -z log n . Also known as zeta function.
References in periodicals archive ?
Limit Theorems for the Riemann Zeta-Function. Kluwer, Dordrecht, 1996.
Mitrovic, The signs of some constants associated with the Riemann zeta-function, Michigan Math.
Ivic, Riemann zeta-function: theory and applications, Oversea Publishing House, 2003.
where [zeta](s) = [[SIGMA].sup.[infinity].sub.k=1][k.sup.-s] (s > 1) is the Riemann zeta-function. Concerning the computational aspects we shall derive an approximation to [gamma](a) in terms of the function [zeta](s, a), assumed to be numerically known.
[6] Ivic, A., 1985, The Riemann Zeta-Function, Dover, New York, NY.
Clearly, [zeta](s, 1) = [zeta](s) and [zeta](s, 2) = ([2.sup.s] - 1) [zeta](s), where [zeta](s) is the Riemann zeta-function. Therefore, in those cases, [zeta](s, [alpha]) is also universal, however, the approximated function f (s) must be non-vanishing on K.
The Riemann zeta-function is a special case of the Hurwitz zeta-function:
The statistical behaviour and universality properties of the Riemann zeta-function and other allied Dirichlet series.
Akatsuka, The Euler product for the Riemann zeta-function in the critical strip.
In [16], Voronin discovered the universality property of the Riemann zeta-function [zeta](s), s = [sigma] + it, on the approximation of analytic functions from a wide class by shifts [zeta](s + i[tau]), [tau] [member of] R.
Where [chi] denotes a Dirichlet character modulo d, L(n, [chi]) denotes the Dirichlet L-function corresponding to[chi], [empty set](d) and [zeta](s) are the Euler function and Riemann zeta-function, respectively.

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