07/17/2014, 05:46 AM
(This post was last modified: 08/02/2014, 02:47 PM by sheldonison.)

(07/14/2014, 12:20 AM)tommy1729 Wrote: .....I'm getting close to a proof that there are no zeros off the real line, all zeros multiplicity 1. It seems to me that the details come to me very slowly, since I keep thinking about it, and keep making steady progress.

This leads to 3 ideas and a remark.

remark : you say " oscillating between positive and negative ".

Now I know that if a closed jordan curve path has all arguments n times then we have n zero's within that closed path.

So that suggests that all real roots have multiplicity 1 and you are in the possesion of a proof ?

Also you have claimed that there are no zero's off the real line ?

Does that have a proof ?

I think it should be provable for Re(z) > 0.

Basically, it comes down to how Kneser's half iterate behaves in the complex plane, along with the definition used for the entire asymptotic function. Interestingly, to understand the negative real axis, you mostly have to understand how Kneser's half iterate behaves for positive , due to the equation . And the definition I'm using for negative axis comes down to plus stuff, where stuff can be shown too small to create zeros anywhere else, since halfk(z) can be shown to have a large absolute value at the negative axis, so stuff winds up acting like a small perturbation of the zeros of the approximation. More later.

- Sheldon