Search results
Results from the WOW.Com Content Network
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.
Similarly Selberg zeta functions satisfy the analogue of the Riemann hypothesis, and are in some ways similar to the Riemann zeta function, having a functional equation and an infinite product expansion analogous to the Euler product expansion. But there are also some major differences; for example, they are not given by Dirichlet series.
The analytic continuation of this zeta function ζ to all complex s ≠ 1; The entire function ξ(s), related to the zeta function through the gamma function (or the Π function, in Riemann's usage) The discrete function J(x) defined for x ≥ 0, which is defined by J(0) = 0 and J(x) jumps by 1/n at each prime power p n. (Riemann calls this ...
The most famous example of a Dirichlet series is = =,whose analytic continuation to (apart from a simple pole at =) is the Riemann zeta function.. Provided that f is real-valued at all natural numbers n, the respective real and imaginary parts of the Dirichlet series F have known formulas where we write +:
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. It is also called the Riemann–Siegel Z function, the Riemann–Siegel zeta function, the Hardy function, the Hardy Z function and the Hardy zeta function.
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large height; Odlyzko, A. (1992), The 10 20-th zero of the Riemann zeta function and 175 million of its neighbors This unpublished book describes the implementation of the algorithm and discusses the results in detail.
Riemann's original use of the explicit formula was to give an exact formula for the number of primes less than a given number. To do this, take F(log(y)) to be y 1/2 /log(y) for 0 ≤ y ≤ x and 0 elsewhere. Then the main term of the sum on the right is the number of primes less than x.