Ad
related to: riemann zeta functional equation
Search results
Results From The WOW.Com Content Network
The equation relates values of the Riemann zeta function at the points s and 1 − s, in particular relating even positive integers with odd negative integers. Owing to the zeros of the sine function, the functional equation implies that ζ ( s ) has a simple zero at each even negative integer s = −2 n , known as the trivial zeros of ζ ( s ) .
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.
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.
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.
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 functional equation in question for the Riemann zeta function takes the simple form = where Z(s) is ζ(s) multiplied by a gamma-factor, involving the gamma function. This is now read as an 'extra' factor in the Euler product for the zeta-function, corresponding to the infinite prime.
Leonhard Euler proved the Euler product formula for the Riemann zeta function in his thesis Variae observationes circa series infinitas (Various Observations about Infinite Series), published by St Petersburg Academy in 1737. [1] [2]
The functional equation () = () is satisfied by the Riemann zeta function, as proved here. The capital Γ denotes the gamma function . The gamma function is the unique solution of the following system of three equations: [ citation needed ]