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  2. 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.

  3. 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.

  4. K-function - Wikipedia

    en.wikipedia.org/wiki/K-function

    Download as PDF; Printable version; In other projects ... where ζ′(z) denotes the derivative of the Riemann zeta function, ζ(a,z) denotes the Hurwitz zeta ...

  5. 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 ...

  6. Wallis product - Wikipedia

    en.wikipedia.org/wiki/Wallis_product

    4 Derivative of the Riemann zeta function at zero. 5 See also. ... Download as PDF; ... The Riemann zeta function and the Dirichlet eta function can be defined: [1]

  7. Prime zeta function - Wikipedia

    en.wikipedia.org/wiki/Prime_zeta_function

    In mathematics, the prime zeta function is an analogue of the Riemann zeta function, studied by Glaisher (1891). It is defined as the following infinite series , which converges for ℜ ( s ) > 1 {\displaystyle \Re (s)>1} :

  8. 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 +:

  9. Bernoulli polynomials - Wikipedia

    en.wikipedia.org/wiki/Bernoulli_polynomials

    They are used for series expansion of functions, and with the Euler–MacLaurin formula. These polynomials occur in the study of many special functions and, in particular, the Riemann zeta function and the Hurwitz zeta function. They are an Appell sequence (i.e. a Sheffer sequence for the ordinary derivative operator).