Riemann Zeta function

From Conservapedia

Jump to: navigation, search

The Riemann Zeta function, commonly denoted ζ(s) is a very important function in number theory. The function can be defined as \zeta(s) = \sum_{n=1}^\infty \frac{1}{n^s} when s has a real part greater than 1. The function can then be analytically continued for everywhere where s is non-one (the function has a simple pole at s = 1). The function was investigated by a number of mathematicians including Euler who showed that \zeta(2) = \frac{\pi^2}{6} but was first extensively investigated by Bernhard Riemann who showed that the behavior of the function was intimately connected to the distribution of prime numbers. In particular, where the function is zero is related very closely to the error term of the Prime Number Theorem and the prime counting function. The Riemann hypothesis is an unsolved conjecture that describes strong restrictions on the locations of the zeros.

The convergence of the Reimann zeta function can be extended over all complex numbers first by stretching to converge over s with real part greater than zero,

\zeta(s) = \left(1-\frac{1}{2^{s-1}}\right)\sum_{n=1}^\infty (-1)^{n+1}\frac{1}{n^s}

and then to all complex numbers via,

\zeta(1-s)=2^{1-s}\pi^{-s}\sin\left(\frac{1-s}{2}\pi\right)\Gamma(s-1)\zeta(s).

Two thing are noticible from the last equation, the function is zero at negative even integers (the trivial zeros) and that the function has dependence on the values on either side of the real 1/2 line.

Retrieved from "http://www.conservapedia.com/Riemann_Zeta_function"
Personal tools