Can someone explain the Riemann hypothesis to me? What is its significance, if any? Will it ever be proved or disproved?
I don’t understand the full idea myself but
http://www.claymath.org/Millennium_Prize_Problems/Riemann_Hypothesis/
gives the very basics.
It’s importance, i believe, is that it can be reformulated into many equivalent statements which are more directlent relevant to fields, such as quantum Physics and.
I think much of modern quantum theory relies on the hypothesis being true, despite it remaining only a hypothesis.
What does it mean when it says * interesting *?
There are other solutions at s = -2, -4, -6, …
These are the so-called trivial solutions. The Riemann Hypothesis says that all other solutions are complex numbers with a real part equal to 1/2 ( the set of all such numbers is known as the critical line in the context of RH). Vast numbers of solutions have been calculated numerically and no counter-example has been found ( obviously, or else the RH would no longer be an open problem). Conrey has proved that at least 40% of the solutions lie on the critical line.
The distribution of the prime numbers is closely related to the zeta function. The usual analytic proof of the Prime Number Theorem, for example, uses one of its comparitively simple properties, namely that [symbol]z/symbol is never 0 when t is real. There is also an “elementary” proof, which is more difficult ( don’t ask).
The Clay Institute site, which puggyfish linked to has a pdf file on the subject, but it is not an easy read.
The May issue of Scientific American had a review of three[!] newly-published books on the Riemann hypothesis. The review is a great intro to the subject.