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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.
English: This is a polar plot of the first 20 non-trivial Riemann zeta function zeros (including Gram points along the critical line (/ +) for real values of running from 0 to 50. The consecutive zeros have 50 red plot points between each with zeros identified by magenta concentric rings (scaled to show the relative distance between their ...
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.
Fonction zêta de Riemann; Usage on gl.wikipedia.org Función zeta de Riemann; Usage on he.wikipedia.org השערת גולדבך החלשה; Usage on id.wikipedia.org Fungsi zeta Riemann; Usage on it.wikipedia.org Funzione zeta di Riemann; Progetto:Laboratorio grafico/Immagini da migliorare/Archivio risolte/77; Usage on lt.wikipedia.org Rymano ...
Description: This image shows the path of the Riemann zeta function along the critical line. That is, it is a graph of (+ /) versus (+ /) for real values of t running from 0 to 34.
These are called its trivial zeros. The zeta function is also zero for other values of s, which are called nontrivial zeros. The Riemann hypothesis is concerned with the locations of these nontrivial zeros, and states that: The real part of every nontrivial zero of the Riemann zeta function is 1 / 2 .
A meromorphic function may have infinitely many zeros and poles. This is the case for the gamma function (see the image in the infobox), which is meromorphic in the whole complex plane, and has a simple pole at every non-positive integer. The Riemann zeta function is also meromorphic in the whole complex plane, with a single pole of order 1 at ...
In 1859 Bernhard Riemann used complex analysis and a special meromorphic function now known as the Riemann zeta function to derive an analytic expression for the number of primes less than or equal to a real number x. Remarkably, the main term in Riemann's formula was exactly the above integral, lending substantial weight to Gauss's conjecture.