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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.
This image shows a plot of the Riemann zeta function along the critical line for real values of t running from 0 to 34. The first five zeros in the critical strip are clearly visible as the place where the spirals pass through the origin. The real part (red) and imaginary part (blue) of the Riemann zeta function along the critical line Re(s) = 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.
This is a polar plot of the first 20 real values r n of the zeta function along the critical line, ζ(1/2 + it), with t running from 0 to 50. 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.
Zeta function of an incidence algebra, a function that maps every interval of a poset to the constant value 1. Despite not resembling a holomorphic function, the special case for the poset of integer divisibility is related as a formal Dirichlet series to the Riemann zeta function.
Riemann zeta function ζ(s) in the complex plane. The color of a point s encodes the value of ζ(s): colors close to black denote values close to zero, while hue encodes the value's argument. In mathematics, analytic number theory is a branch of number theory that uses methods from mathematical analysis to solve problems about the integers. [1]
where ζ(s) is the Riemann zeta function (which is undefined for s = 1). The multiplicities of distinct prime factors of X are independent random variables. The Riemann zeta function being the sum of all terms for positive integer k, it appears thus as the normalization of the Zipf distribution. The terms "Zipf distribution" and the "zeta ...
A prototypical example, the Riemann zeta function has a functional equation relating its value at the complex number s with its value at 1 − s. In every case this relates to some value ζ(s) that is only defined by analytic continuation from the infinite series definition.