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Zeros of the Riemann zeta except negative even integers are called "nontrivial zeros". The Riemann hypothesis states that the real part of every nontrivial zero must be 1 / 2 . In other words, all known nontrivial zeros of the Riemann zeta are of the form z = 1 / 2 + yi where y is a real number.
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.
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
A Gram point is a point on the critical line 1/2 + it where the zeta function is real and non-zero. Using the expression for the zeta function on the critical line, ζ(1/2 + it) = Z(t)e −iθ(t), where Hardy's function, Z, is real for real t, and θ is the Riemann–Siegel theta function, we see that zeta is real when sin(θ(t)) = 0.
This sum conjecture is also known as Sum Theorem, and it may be expressed as follows: the Riemann zeta value of an integer n ≥ 2 is equal to the sum of all the valid (i.e. with s 1 > 1) MZVs of the partitions of length k and weight n, with 1 ≤ k ≤ n − 1. In formula: [3]
It is also possible to argue for the value of − + 1 / 12 ... including ζ(−1) = − + 1 / 12 . In zeta function regularization, the series ...
Gut, Allan. "Some remarks on the Riemann zeta distribution". CiteSeerX 10.1.1.66.3284. What Gut calls the "Riemann zeta distribution" is actually the probability distribution of −log X, where X is a random variable with what this article calls the zeta distribution. Weisstein, Eric W. "Zipf Distribution". MathWorld
where ζ is the Riemann zeta function. It has an approximate value of [1] ζ(3) ≈ 1.20205 69031 59594 28539 97381 61511 44999 07649 86292 … (sequence A002117 in the OEIS). It is named after Roger Apéry, who proved that it is an irrational number.