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The Riemann zeta function ζ(z) plotted with domain coloring. [1] The pole at = and two zeros on the critical line.. The Riemann zeta function or Euler–Riemann zeta function, denoted by the Greek letter ζ (), is a mathematical function of a complex variable defined as () = = = + + + for >, and its analytic continuation elsewhere.
Gourdon (2004), The 10 13 first zeros of the Riemann Zeta function, and zeros computation at very large height; Odlyzko, A. (1992), The 10 20-th zero of the Riemann zeta function and 175 million of its neighbors This unpublished book describes the implementation of the algorithm and discusses the results in detail.
Z function in the complex plane, plotted with a variant of domain coloring. Z function in the complex plane, zoomed out. In mathematics, the Z function is a function used for studying the Riemann zeta function along the critical line where the argument is one-half.
Since the Hurwitz zeta function is a generalization of the Riemann zeta function, we have γ n (1)=γ n The zeroth constant is simply the digamma-function γ 0 (a)=-Ψ(a), [28] while other constants are not known to be reducible to any elementary or classical function of analysis. Nevertheless, there are numerous representations for them.
The zeta function values listed below include function values at the negative even numbers (s = −2, −4, etc.), for which ζ(s) = 0 and which make up the so-called trivial zeros. The Riemann zeta function article includes a colour plot illustrating how the function varies over a continuous rectangular region of the complex plane.
Zeta function of an incidence algebra, a function that maps every interval of a poset to the constant value 1. Despite not resembling a holomorphic function, the special case for the poset of integer divisibility is related as a formal Dirichlet series to the Riemann zeta function.
In mathematics, a rational zeta series is the representation of an arbitrary real number in terms of a series consisting of rational numbers and the Riemann zeta function or the Hurwitz zeta function. Specifically, given a real number x, the rational zeta series for x is given by = = (,)
Zeta functions and L-functions express important relations between the geometry of Riemann surfaces, number theory and dynamical systems.Zeta functions, and their generalizations such as the Selberg class S, are conjectured to have various important properties, including generalizations of the Riemann hypothesis and various relationships with automorphic forms as well as to the representations ...