Challenges in Proving Zeros of Zeta Function: Lack of Tools or Creativity?

[jacksonwalter]

What about the Riemann Zeta function makes it so difficult to prove that all the zeros have real part 1/2? Is it that we lack the discoveries and tools necessary, or we just aren't creative enough, or maybe both? Same question for Goldbach's. Fermat's seemed to rely on elliptic curves which have really only been invented/discovered rather recently in relation the time the problem was posed.

 

[CRGreathouse]

What about the Riemann Zeta function makes it so difficult to prove that all the zeros have real part 1/2? Is it that we lack the discoveries and tools necessary, or we just aren't creative enough, or maybe both? Same question for Goldbach's. Fermat's seemed to rely on elliptic curves which have really only been invented/discovered rather recently in relation the time the problem was posed.

We're making progress on the Riemann hypothesis, but there's still a lot of work to go. Similarly, Goldbach's conjecture has had great progress -- we've nearly proved the weak version (only finitely many verifications to go!).

In that sense they're unlike the Collatz conjecture where we appear to lack the tools to attack the problem.

 

[Dragonfall]

Re: Goldbach, do you mean Chen-Jing Run's theorem?

 

[jacksonwalter]

We're making progress on the Riemann hypothesis, but there's still a lot of work to go. Similarly, Goldbach's conjecture has had great progress -- we've nearly proved the weak version (only finitely many verifications to go!).

In that sense they're unlike the Collatz conjecture where we appear to lack the tools to attack the problem.

Sweet, thanks.

So what consequences will occur as a result of proving the Riemann Hypothesis?

 

[CRGreathouse]

So what consequences will occur as a result of proving the Riemann Hypothesis?

There are many, but the most important one to me is the tight bound on pi(x) due to Schoenfeld.

 

[jacksonwalter]

There are many, but the most important one to me is the tight bound on pi(x) due to Schoenfeld.

So what exactly do you mean by us having made great progress? It's hard to see how you can 'almost' have proved something. Are there any specific stumbling blocks or general properties of the Riemann Hypothesis that make it in particular especially difficult to solve?

 

[CRGreathouse]

So what exactly do you mean by us having made great progress?

First we proved that all nontrivial zeros were on the critical strip, then that they were all on the strict critical strip (none on the 'edges'), then that infinitely many were on the critical line, then that a positive fraction were on the critical line.

There are others things, of course, but that's the main thrust of progress.