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  2. Proof of the Euler product formula for the Riemann zeta ...

    en.wikipedia.org/wiki/Proof_of_the_Euler_product...

    The method of Eratosthenes used to sieve out prime numbers is employed in this proof. This sketch of a proof makes use of simple algebra only. This was the method by which Euler originally discovered the formula. There is a certain sieving property that we can use to our advantage:

  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. Euler product - Wikipedia

    en.wikipedia.org/wiki/Euler_product

    where ω(n) counts the number of distinct prime factors of n, and 2 ω(n) is the number of square-free divisors. If χ ( n ) is a Dirichlet character of conductor N , so that χ is totally multiplicative and χ ( n ) only depends on n mod N , and χ ( n ) = 0 if n is not coprime to N , then

  5. 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.

  6. 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.

  7. Functional equation (L-function) - Wikipedia

    en.wikipedia.org/wiki/Functional_equation_(L...

    The functional equation in question for the Riemann zeta function takes the simple form = where Z(s) is ζ(s) multiplied by a gamma-factor, involving the gamma function. This is now read as an 'extra' factor in the Euler product for the zeta-function, corresponding to the infinite prime.

  8. Category:Zeta and L-functions - Wikipedia

    en.wikipedia.org/wiki/Category:Zeta_and_L-functions

    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 ...

  9. 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.