# Riemann zeta function

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**Riemann zeta function**— ζ(s) in the complex plane. The color of a point s encodes the value of ζ(s): dark colors denote values close to zero and hue encodes the value s argument. The white spot at s = 1 is the pole of the zeta function; the black spots on the… … Wikipedia**Riemann zeta function**— ▪ mathematics function useful in number theory for investigating properties of prime numbers (prime). Written as ζ(x), it was originally defined as the infinite series ζ(x) = 1 + 2−x + 3−x + 4−x + ⋯. When x = 1, this series is called the… … Universalium**Riemann zeta-function**— noun A particular function, whose domain is the set of complex numbers (except 1), and whose codomain is the set of complex numbers … Wiktionary**Proof of the Euler product formula for the Riemann zeta function**— We will prove that the following formula holds::egin{align} zeta(s) = 1+frac{1}{2^s}+frac{1}{3^s}+frac{1}{4^s}+frac{1}{5^s}+ cdots = prod {p} frac{1}{1 p^{ s end{align}where zeta; denotes the Riemann zeta function and the product extends over… … Wikipedia**Riemann Xi function**— In mathematics, the Riemann Xi function is a variant of the Riemann zeta function, and is defined so as to have a particularly simple functional equation. The function is named in honour of Bernhard Riemann.DefinitionRiemann s lower case xi is… … Wikipedia**Zeta function universality**— In mathematics, the universality of zeta functions is the remarkable property of the Riemann zeta function and other, similar, functions, such as the Dirichlet L functions, to approximate arbitrary non vanishing holomorphic functions arbitrarily… … Wikipedia**Zeta function**— A zeta function is a function which is composed of an infinite sum of powers, that is, which may be written as a Dirichlet series::zeta(s) = sum {k=1}^{infty}f(k)^s Examples There are a number of mathematical functions with the name zeta function … Wikipedia**Zeta function regularization**— In mathematics and theoretical physics, zeta function regularization is a type of regularization or summability method that assigns finite values to superficially divergent sums. The technique is now commonly applied to problems in physics, but… … Wikipedia**Hurwitz zeta function**— In mathematics, the Hurwitz zeta function, named after Adolf Hurwitz, is one of the many zeta functions. It is formally defined for complex arguments s with Re( s )>1 and q with Re( q )>0 by:zeta(s,q) = sum {n=0}^infty frac{1}{(q+n)^{sThis series … Wikipedia**Dedekind zeta function**— In mathematics, the Dedekind zeta function of an algebraic number field K, generally denoted ζK(s), is a generalization of the Riemann zeta function which is obtained by specializing to the case where K is the rational numbers Q. In particular,… … Wikipedia