|Why is Riemann's Hypothesis so important?|
The verification of Riemann's Hypothesis (formulated in 1859)
is considered to be one of modern mathematic's most important problems.
The last 140 years did not bring its proof, but a considerable number of important mathematical theorems which depend on the Hypothesis
being true, e.g. the fastest known primality test of Miller.
The Riemann zeta function is defined for Re(s)>1 by
and is extended to the rest of the complex plane (except for s=1) by analytic continuation.
The Riemann Hypothesis asserts that all nontrivial zeros of the zeta function are on the critical line (1/2+it where t is a real number).
To verify empirically the Riemann Hypothesis for certain regions
and make it usable, in 1903
the first fifteen zeros of Riemann's zeta function
on the critical line were calculated. Thus, the Riemann Hypothesis is true at least in the region |t| < 65.801.
|Participate in the verification of Riemann's Hypothesis!|
Today, we have better resources to verify or falsify Riemann's Hypothesis. First
the high-speed computers, then the networks have increased the capacity of calculations.
Now we want to go one step further by bundling up the resources into a grid network.
Therefore, I invite all interested people to participate in the verification of the zeros
of the Riemann zeta function for a new record.
Before I have started with the computation on August 28, 2001, the hypothesis has been checked for the first 1,500,000,001 zeros.
On October 27, 2001, J. van de Lune checked the hypothesis for the first 10 billion zeros.
Up to now, it has been extended to the first 100 billion zeros which required more than
1.3×1018 floating-point operations.
But we will go beyond it by the top producers.
|What have we achieved so far?|
The result of the computation which verified the first 100 billion zeros
of the Riemann zeta function confirms the calculations made previously by
and extends these to the first 100 billion nontrivial zeros. All these zeros
of the form + it have real part
= 1/2 and are simple. Thus, the Riemann
Hypothesis is true at least for all |t| < 29,538,618,432.236.
More details about this computation will soon appear in a paper.