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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. Particular values of the Riemann zeta function - Wikipedia

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

    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.

  4. Riemann–Siegel formula - Wikipedia

    en.wikipedia.org/wiki/Riemann–Siegel_formula

    Siegel derived it from the Riemann–Siegel integral formula, an expression for the zeta function involving contour integrals. It is often used to compute values of the Riemann–Siegel formula, sometimes in combination with the Odlyzko–Schönhage algorithm which speeds it up considerably.

  5. 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large height; Odlyzko, A. (1992), The 10 20-th zero of the Riemann zeta function and 175 million of its neighbors This unpublished book describes the implementation of the algorithm and discusses the results in detail.

  6. Analytic number theory - Wikipedia

    en.wikipedia.org/wiki/Analytic_number_theory

    Riemann zeta function ζ(s) in the complex plane. The color of a point s encodes the value of ζ ( s ): colors close to black denote values close to zero, while hue encodes the value's argument . In mathematics , analytic number theory is a branch of number theory that uses methods from mathematical analysis to solve problems about the integers ...

  7. On the Number of Primes Less Than a Given Magnitude

    en.wikipedia.org/wiki/On_the_Number_of_Primes...

    The analytic continuation of this zeta function ζ to all complex s ≠ 1; The entire function ξ(s), related to the zeta function through the gamma function (or the Π function, in Riemann's usage) The discrete function J(x) defined for x ≥ 0, which is defined by J(0) = 0 and J(x) jumps by 1/n at each prime power p n. (Riemann calls this ...

  8. 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 ...

  9. Rational zeta series - Wikipedia

    en.wikipedia.org/wiki/Rational_zeta_series

    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 = = (,) where each q n is a rational number, the value m is held ...