A $1,000,000 Question

shmoe

Sure, it's a rule that determines where the primes are, but remember one of the goals of number theorists is to improve the error term in the prime number theorem. Sieves and other elementary methods (essentially meaning no complex analysis) have produced some results, but nothing like we can do with the zeta function as far as the error term goes.

Yes, zeros and primes are the same thing in a sense, so if you find out what's up with one of them you know about the other. In this sense this dynamical system would tell you about the primes, and can be thought of the thing controlling them (though you could probably think of the primes as controlling the dynamical system), so you could think of it as 'the same thing' as what you've described in a way.

However, if you ran into this mythical dynamical system in a back alley it would probably be impossible to draw a connection between it and primes without the zeta function in hand to translate back and forth. This is the sense that I mean the dynamical system is more naturally attatched to zeta and not the primes.

Canute

Hmm. I still can't understand why this system is said to be mysterious, but never mind.

I think I need to stop here and go away to think some things through, now you've explained some of the mathematics to me. Thanks for all your help and patience. I'll put the rest of my thoughts in better order and may be back to ask you some more questions. I can at least now see the direction I need to head in.

Many thanks
Canute

shmoe

They don't even know what the dynamical system they are looking for is, let alone if it even exists, that is the mystery. If they could find this system like they hope, the riemann hypothesis would be solved, that is how transparent this hypothetical dynamical system is expected to make things.

You are welcome.