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phtran

Riemann hypothesis solved?

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Proof of Riemann hypothesis on one page. Who would be excited if RH was proven today?

Riemann.png

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Posted (edited)
On 6/26/2019 at 7:01 AM, phtran said:

Proof of Riemann hypothesis on one page. Who would be excited if RH was proven today?

 

I'm not taking you seriously 100 %. But I'm interested in your definitions.

1) Your notation is confusing at best. I would first drop the h bar. I don't know what quantum mechanics is doing there. Also, no need for the partial derivative symbol.

\[P=-i\frac{d}{dx}\]

Your left and right shift operators I have re-named L and R:

\[f_{R}\left(x\right)=Rf=f\left(x\right)-f\left(x+1\right)\]

\[f_{L}\left(x\right)=Lf=f\left(x\right)-f\left(x-1\right)\]

So that,

\[G=L\left(XP+PX\right)R\]

I'm not sure your operator is compact or admits a compact extension.

2) It seems like you're extending Riemann's zeta function to include a real variable x taking values on the real line.  Taking x=1 doesn't take you back to Riemann's zeta function. You should be more explicit about what you're doing there.

3) Your definition of scalar product involves only integration/sum from 1 to positive infinity. That would require some heavy-duty extensions and checks of your definitions (closeness, domains, etc.)

4) X and P are unbounded operators that do not belong to the trace class operators. IOW: Look out for mistakes, especially if you're using anything like traces. Example of simple arguments that miserably fail with them:

\[\textrm{tr}\left(XP-PX\right)=0\]

\[\textrm{tr}(-iI)=-i\times\infty\]

5) Soundness of passing an operator from left to right in a scalar product depends on subtle questions about domains, not only on real character of its "formal" spectrum. You must check that you're not letting out the accumulation points of your domain in case it's not compact (topological argument.)

6) Related to the previous: Your "left operator" takes you out of the domain of your scalar product. Doesn't that have any bearing on your "proof" of something?

I'm just curious. That's all.

Edited by joigus
bad rendering of eq.

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Posted (edited)

Excellent reply +1. I don't have much too add as of yet. 

Edited by Mordred

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15 hours ago, Mordred said:

Excellent reply +1. I don't have much too add as of yet. 

Thank you. I'm sure any comments you may have to add will be most interesting.

18 hours ago, joigus said:

must check that you're not letting out the accumulation points of your domain

Here I meant domain and spectrum.

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