An strategy to prove Riemann hypothesis

[eljose]

Let be the Hamitonian of a particle with mass m in the form:

$$H=\frac{-\hbar^{2}}{2m}D^{2}\phi(x)+V(x)\phi(x)$$

then the RH is equivalent to prove that exist a real potential V(x) of the Hamiltonian so that the values E_n $$H\phi=E_{n}\phi$$ satisfy the equation $$\zeta(1/2+iE_{n})=0$$ that is the roots of the Riemann zeta function are the Energies of the system...

you only need to prove that V(x) is real,so the H will be self-adjoint,we can write the solution of the problem (approximately) by the wave function:

$$\phi=e^{iS/\hbar}$$ with $$S=\int(2mE_{n}-2mV(x))^{0.5}dx$$

with that you can substituting into Schroedinguer equation get a differential equation of second order for V(x) $$F(x,V(x),DV(x),D^{2}V(x),E_{n})=0$$


the key is that you needn,t solve the equation you only have to prove that the potential will be real by knowing that energies satisfy $$\zeta(1/2+iE_{n})=0$$ i will try to submit to some math teacher to see if i can do my PhD in math-physics in this matter...

 

[Hurkyl]

This is completely unclear. I have absolutely no idea how you intend to try and prove the Riemann hypothesis. The only thing I can gather is that, for some reason, you want to try and construct a differential operator of a particular form whose eigenvalues are related somehow to ζ.

Could you try again, sketching an outline of the proof that you hope to be able to complete?

 

[matt grime]

Actually eljose has, perhaps (it is as ever unlcear), somehow stumbled on something that is acutally of interest. it is believed that random matrix theory might play some role in working towards a substantial part of a possible proof of RH. see the work of Berry, Keating, Mezzadri, Snaith et al.

 

[eljose]

To hurkyl:i intend to prove in a thesis that the roots of the operators could (or should) be the eigenvalues of a certain self-adjoint operator so they will be real..the RH is similar to prove that the roots of the function $$\zeta(1/2+it)$$ are all real...the strategy is to prove that will exist a real potential V(x) for a given Hamiltonian in the sense that the eigenvalues of this Hamiltonian will be the values of the root of the function $$\zeta(1/2+it)$$ then i state an approximate differential equation for the potential V,the main key is to show that this potential will be real..

 

[eljose]

In fact RH can be easily proved by this...if $$\zeta(1/2+is)$$ is a root also $$\zeta(1/2-is^*)=\zeta(1/2+is*)=0$$ so if s is an energy also s* will be another energy so the potential is real $$<\phi*_{n}|T*+V*|\phi*_{n}>=<\phi_{k}|T+V|\phi_{k}>$$ from this we deduce V is real...

alternatively we can prove for $$\zeta(a+is)$$ that if s is a root also s*+(2a-1)i must be a root so there are complex energies,if this happens the potential must be complex, by the same argument than above a complex potential can not have real energies,so there are no real roots for the zeta function of the form $$\zeta(a+is)$$ with a different from 1/2