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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.
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 .
The Riemann hypothesis is a conjecture about the distribution of the zeros of the Riemann zeta-function ζ(s). The Riemann zeta-function is defined for all complex numbers s ≠ 1. It has zeros at the negative even integers (i.e. at s=-2, s=-4, s=-6, ...). These are called the trivial zeros. The Riemann hypothesis is concerned with the non ...
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).
In 1914, Godfrey Harold Hardy proved [1] that the Riemann zeta function (+) has infinitely many real zeros. Let () be the total number of real zeros, () be the total number of zeros of odd order of the function (+), lying on the interval (,].
Z function in the complex plane, plotted with a variant of domain coloring. Z function in the complex plane, zoomed out. In mathematics, the Z function is a function used for studying the Riemann zeta function along the critical line where the argument is one-half.
Since the Hurwitz zeta function is a generalization of the Riemann zeta function, we have γ n (1)=γ n The zeroth constant is simply the digamma-function γ 0 (a)=-Ψ(a), [28] while other constants are not known to be reducible to any elementary or classical function of analysis. Nevertheless, there are numerous representations for them.