The Riemann Zeta-function z(s)
(pdf version : pdf;
(postscript version : ps)
The Riemann Zeta-function is defined as
for complex values of s. While converging only for complex numbers
s with Â(s) > 1 , this function can be analytically continued to
the whole complex plane (with a single pole at s=1).
The Riemann Zeta-function was first introduced by Euler with the
computation of
but it was Riemann who, in the 1850's, generalized its use and showed
that the distribution of primes is related to the location of the
zeros of z(s). Riemann conjectured that the non trivial zeros of
z(s) are located on the critical line Â(s)=1/2. This
conjecture, known as the Riemann hypothesis, has never been
proved or disproved, and is probably the most important unsolved
problem in mathematics. The Riemann hypothesis makes the zeta function
so famous, and numerical computation have been made to check it for
various sets of zeros.
We expose here the most classical results about the zeta-function,
together with some computational aspects. Our presentation is divided
into several parts that are listed here :
Related links on Riemann Zeta function
Back to
numbers,
constants and computation
File translated from
TEX
by
TTH,
version 3.40.
On 14 Dec 2004, 14:31.
Computation of zeros of the Zeta function.
Contains announcement and details of the latest record about Riemann
Hypothesis numerical verification (on the first 1013 zeros),
and about zeros computation at very large height.