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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.
The Riemann zeta function is defined for other complex values via analytic continuation of the function defined for σ > 1. Leonhard Euler considered the above series in 1740 for positive integer values of s , and later Chebyshev extended the definition to Re ( s ) > 1. {\displaystyle \operatorname {Re} (s)>1.} [ 4 ]
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. Zeta function of an operator or spectral zeta function
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.
When all of the are n th roots of unity and the are all nonnegative integers, the values of the multiple polylogarithm are called colored multiple zeta values of level. In particular, when n = 2 {\displaystyle n=2} , they are called Euler sums or alternating multiple zeta values , and when n = 1 {\displaystyle n=1} they are simply called ...
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 k − s {\displaystyle k^{-s}} for positive integer k , it appears thus as the normalization of the Zipf distribution .
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The Bernoulli numbers can be expressed in terms of the Riemann zeta function as B n = −nζ(1 − n) for integers n ≥ 0 provided for n = 0 the expression −nζ(1 − n) is understood as the limiting value and the convention B 1 = 1 / 2 is used. This intimately relates them to the values of the zeta function at negative integers.