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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 .
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.
The Riemann zeta function is also meromorphic in the whole complex plane, with a single pole of order 1 at z = 1. Its zeros in the left halfplane are all the negative even integers, and the Riemann hypothesis is the conjecture that all other zeros are along Re(z) = 1/2.
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
The zeros of the eta function include all the zeros of the zeta function: the negative even integers (real equidistant simple zeros); the zeros along the critical line, none of which are known to be multiple and over 40% of which have been proven to be simple, and the hypothetical zeros in the critical strip but not on the critical line, which if they do exist must occur at the vertices of ...
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large height; Odlyzko, A. (1992), The 10 20-th zero of the Riemann zeta function and 175 million of its neighbors This unpublished book describes the implementation of the algorithm and discusses the results in detail.