This involves the zeta function
Today I was speaking to a friend of mine, and he brought up something called Riemann's Hypothesis. He told me that it was a concept relating to the distribution of primes and has yet to be proven, and is evidently somehow related to modern financial security systems.
However, upon researching the topic I've found very little in the area of practical application of the Hypothesis... and add to the fact I find the hypothesis confusing to understand.
Would anyone care to explain the hypothesis to me in layman's terms, and what it means holistically speaking?
The simply is no "lay" explaination for this particular hypothesis. The very nature of the conjecture requires that you know someting of several different areas of advanced mathematics. I'm not too bad at mathematics, and I've tried to interpret the meaning with no success.
The Riemann Zeta function is defined by
Note that is defined only for with .
One can extend by noticing (this can be shown fairly easily). This is called an extension because now is defined for (except for where there is a simple pole).
The Riemann Hypothesis states that the only zeros of in the strip have real part equal to i.e. .
I'd be happy to explain.
We want to know when converges. For this we'll need a couple facts about complex numbers.
So . Which is convergent for by the p-test.
Thus is absolutely convergent for .
Hi. I think there is a layman's way of explaining it. Start with:
Alright, you're insulted right now. How then about: . Same dif I know. Well, we can also consider integral functions like:
Then when is that one zero? Well, piece of cake. Integrate it, it's a quadratic in x, then find the roots via the quadratic formula. Well, then, that's what the Riemann Hypothesis is about except the integral function is more complicated and it deals with complex numbers:
Now, for which values of is that one zero? The Riemann Hypothesis claims that in the half-plane, , all of them have the form
"I've found very little in the area of practical application of the Hypothesis..."
A simple check by Googling would confirm that (my attitude is that if it isn't on the internet, it doesn't exist). When you say "practical application", do you mean within math or outside of it?
"...and is evidently somehow related to modern financial security systems."
That would be because of the primes.
It would help to know something of your math background to try to answer your questions.