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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.
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.
The values of r n in this range are the first 10 non-trivial Riemann zeta function zeros and the first 10 Gram points, each labeled by n. Fifty red points have been plotted between each r n, and the zeros are projected onto concentric magenta rings scaled to show the relative distance between their values of t. Gram's law states that the curve ...
The real line describes the two-point correlation function of the random matrix of type GUE. Blue dots describe the normalized spacings of the non-trivial zeros of Riemann zeta function, the first 10 5 zeros. In the 1980s, motivated by Montgomery's conjecture, Odlyzko began an intensive numerical study of the statistics of the zeros of ζ(s).
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large height; Odlyzko, A. (1992), The 10 20-th zero of the Riemann zeta function and 175 million of its neighbors This unpublished book describes the implementation of the algorithm and discusses the results in detail.
In mathematics, the Riemann–von Mangoldt formula, named for Bernhard Riemann and Hans Carl Friedrich von Mangoldt, describes the distribution of the zeros of the Riemann zeta function. The formula states that the number N(T) of zeros of the zeta function with imaginary part greater than 0 and less than or equal to T satisfies
In the study of the Riemann hypothesis, a Lehmer pair is a pair of zeros of the Riemann zeta function that are unusually close to each other. [1] They are named after Derrick Henry Lehmer, who discovered the pair of zeros
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 ...