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  2. Riemann zeta function - Wikipedia

    en.wikipedia.org/wiki/Riemann_zeta_function

    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.

  3. Riemann hypothesis - Wikipedia

    en.wikipedia.org/wiki/Riemann_hypothesis

    Similarly Selberg zeta functions satisfy the analogue of the Riemann hypothesis, and are in some ways similar to the Riemann zeta function, having a functional equation and an infinite product expansion analogous to the Euler product expansion. But there are also some major differences; for example, they are not given by Dirichlet series.

  4. Particular values of the Riemann zeta function - Wikipedia

    en.wikipedia.org/wiki/Particular_values_of_the...

    Zeros of the Riemann zeta except negative even integers are called "nontrivial zeros". 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.

  5. Dirichlet series - Wikipedia

    en.wikipedia.org/wiki/Dirichlet_series

    The most famous example of a Dirichlet series is = =,whose analytic continuation to (apart from a simple pole at =) is the Riemann zeta function.. Provided that f is real-valued at all natural numbers n, the respective real and imaginary parts of the Dirichlet series F have known formulas where we write +:

  6. Zeta distribution - Wikipedia

    en.wikipedia.org/wiki/Zeta_distribution

    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 ...

  7. Li's criterion - Wikipedia

    en.wikipedia.org/wiki/Li's_criterion

    The Riemann ξ function is given by = / ()where ζ is the Riemann zeta function.Consider the sequence = ()! [⁡ ()] | =. Li's criterion is then the statement that the Riemann hypothesis is equivalent to the statement that > for every positive integer .

  8. Bernoulli number - Wikipedia

    en.wikipedia.org/wiki/Bernoulli_number

    The Bernoulli numbers can be expressed in terms of the Riemann zeta function as B n = −nζ(1 − n) for integers n ≥ 0 provided for n = 0 the expression −nζ(1 − n) is understood as the limiting value and the convention B 1 = ⁠ 1 / 2 ⁠ is used. This intimately relates them to the values of the zeta function at negative integers.

  9. Odlyzko–Schönhage algorithm - Wikipedia

    en.wikipedia.org/wiki/Odlyzko–Schönhage_algorithm

    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.