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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.
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 .
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. It is also called the Riemann–Siegel Z function, the Riemann–Siegel zeta function, the Hardy function, the Hardy Z function and the Hardy zeta function.
The Riemann zeta function along the critical line, all complex numbers with a real part of a half. That is, it is a graph of ((+ /)) versus ((+ /)) for real values of t running from 0 to 34. The first five zeros in the critical strip are clearly visible as the place where the spirals pass through the origin.
A regular finite graph is a Ramanujan graph, a mathematical model of efficient communication networks, if and only if its Ihara zeta function satisfies the analogue of the Riemann hypothesis as was pointed out by T. Sunada.
where ζ(s) is the Riemann zeta function (which is undefined for s = 1). The multiplicities of distinct prime factors of X are independent random variables. The Riemann zeta function being the sum of all terms for positive integer k, it appears thus as the normalization of the Zipf distribution. The terms "Zipf distribution" and the "zeta ...
Interactive maps, databases and real-time graphics from The Huffington Post
Duursma zeta function of error-correcting codes; Epstein zeta function of a quadratic form; Goss zeta function of a function field; Hasse–Weil zeta function of a variety; Height zeta function of a variety; Hurwitz zeta function, a generalization of the Riemann zeta function; Igusa zeta function; Ihara zeta function of a graph; L-function, a ...