RH involves limits of mobius transformations

[Jonny_trigonometry]

I was wondering if one of the approaches to proving the RH involves limits of mobius transformations of the zeta function on the right side of the imaginary axis such that all the non-trivial zeros get mapped into an annulus; this annulus is then shown to contain an infinite number of zeros via the argument principle in the lim as the annulus' radii converge to the same value. In the lim, you have a mobius transformation of the critical line to the circumferance of the unit circle.

 

[shmoe]

I don't see the point of this transformation when you can work with the critical strip just as easily.

Also, we already know there are infinitely many zeros on the critical line.

 

[Jonny_trigonometry]

hmm... The second part of the proof involves showing that there are no zeros on the right and left sides of the critical strip (on the right side of the imaginary axis), i was thinking that a transformation would let you use the argument principle. Eh, obviously I don't have any idea how to approach the RH.

 

[shmoe]

The argument principle is used to count zeros in the strip, using a rectangle whose vertical sides lie outside the strip itself. You run into troubles if you try to move this inside the strip as you need bounds for $$\zeta'(s)/\zeta(s)$$ on your rectangle. You're only going to be able to do this if you've already proven there's no zeros here, which kinda defeats the purpose.

You can be quite accurate with the argument principle to count roots. Accurate enough that you can determine exactly the number of roots up to a given height, then if you can find that number of roots on the critical line you can prove RH up to this height.

 

[Jonny_trigonometry]

Very interesting. From what I knew, I thought the argument principle works only for inside the unit circle... I've never really thought about how to analyze the critical line until I took complex analysis. It just beckons a person to try and prove the RH for a rectangle increasingly tall, then again, that is why it's an attractive problem, and its elusiveness is even more attractive.

 

[shmoe]

Very interesting. From what I knew, I thought the argument principle works only for inside the unit circle...

Often in complex analysis you prove something for the simpler case of circles centered at the origin then apply some transformation that takes these cricles to other regions.

You might want to look up the "Riemann-von Mangoldt formula" to see how they count the zeros in the strip.