A proof of RH using quantum physics

[matt grime]

I do not deny that given a set of complex numbers satisfying certain properties that one can construct a potential with these and only these as the as its spectrum. HOwever, I have yet to see any stated reason why this will work for the zeta function, and that from it one can clconlude the Riemann hypothesis. approximate solutions are not accepetable.


opinion of what is and isn't rigorous is completely irrelevant, jose. since you are claiming to do maths you must accpet the standards of mathematics. if we were taking "suggestive" reasons then of course RH is true since we know it is emprically true for such a ragne of s that it cannot help but be true for all s. but that is different from proving it.

you are the person who keeps claiming to have a proof. if you dont' like what that really entails tehn stop abusign the word.

 

[eljose]

why is not my method acceptable?..from the integral equation above you could construct the potential V(x) by solving a non-linear equation for numerical values V(xj) j=1,2,3...n or for example we could use an existence theorem for integral equations so we can deduce that the potential exists...wouldn,t that be acceptable?. we have the system of equations for the Potential in the form:

\sum_{j}K(n,V(x_{j}))=g(n) now set n=1,2,3...k so we have the roots E1,E2,E3,...Ek upto afinite number k (we know the roots of Riemann function upto 10^{12} or even more.

I have a proof of RH the only "inconsistency" according to you is that i have not proved that the potential exist...but here i am giving an example of how to obtain its numerical value,now with those numerical value we could "modelize" our potential...

The other chance is to use the existence theorem of integral equations for the Kernel K(n,V(x)) and show that this integral exist..a question on rigour,if you prove that can calculate numerical values of the potential wouldn,t you have proved this potential exists?...

 

[matt grime]

your method isn't acceptable because it doesn't prove anything. i can tell that by the way you haven't proven anything; bit of a give away. all i see is a load of:

if i do this, then i might be able to do something, that might up to an error term do something else, but i can't prove any of it.

that isn't a proof.


given the zeroes, I'm sure you can construct some differential operator as required, however even if that were the case then you've not shown how one may determine that it is hermitian.

to do that one would need to make V(x) a function of zeta. zeta has not appeared at any point in your calculations.

since you are not assuming that the non trivial zeroes lie on the critical line you are somehow creating a hermitian operator from these values. since zeta has not been mentioned at all in your working then the same logic applies to any complex numbers, and indeed one can conclude, accoridng to your logic, that all complex numbers are in fact real.

you see, I don't need to show that something cannot be done (since it might be doable) to show that a method given is nonsense. as yours is as best we can tell.

example: let S be the set of zeroes of the function

sin(iz)

create a potential according to our method, and from this i conclude that all zeroes of this must satisfy z=z* ie are real since the operator so created is hermitian. however, I know where the zeroes are, thank you, and they are certainly not at real numbers, they are at purely imaginary numbers.

so, why doesn't your method apply here? i see no reason for it not to.

 

[EL]

Credit to matt grime for his patient arguing with a wall!

 

[matt grime]

And I further see no reason why your method picks out exactly the non-trivial zeroes and nothing else. but then the dependence on zeta has never been explained.

so, if you want to make this metho work:

take g(z) some function frmo C to C.

explain how we may take a restricted set of its zeroes, call them T, and from T create a differential operator

L=\partial^2_x + V_g

such that the spectrum is exactly T

now specify the conditions on g and T that allow us to conclude L is hermitian, and hence T must be a set of real numbers.

You have not in any decipherable way done any of those things.

I am perfectly willing to believe that it can be done. but I have not seen you do it in a way that convinces me at all.

 

[eljose]

let be perturbation theory:(at first order of the energies)...

E_{n}-E^{0}_{n}=\delta{E(n)}=<\phi|V|\phi> (1)

where E_{n} are the "energies" roots of the Riemann Z(1/2+is) and E^0_{n} are the Eneriges of the Hamiltonian H0=P^{2}/2m then (1) provides an integral equation for V we could find a resolvent kernel R for this so:

V(x)=\int_{-\infty}^{\infty}dnR(n,x)\delta{E(n)}

with the resolvent Kernel: R=\sum_{m=0}^{\infty}b_{m}(K-cI)^{m} is the Taylor expansion of the operator K^{-1} and K^{m} is the m-th iterated kernel,I is the identity operator and c is a real constant so the series converge for ||K||<c

\delta{E(n)}=K[V(x)]

c is a constant so the series converges,the radius of convergence is given by the norm of K operator ||K||<c

we have proved that V exist at first order in perturbation theory...

with K the kernel given by K(x,n)=|\phi(x,n)|^{2}
phi are the eigenfunctions of the Hamiltonian H0 epending on n i have applied Neumann series to the operator to obtain the resolvent Kernel...

 

[David]

I'll let Matt continue to point out the circularity of the argument being presented here adn tackle it from a technical viewpoint but will add one other little piece of information to the discussion, just to demonstrate the futility of this in a completely different way.

Supposing this were a valid proof, then credit for it should go to other people who have discovered it long ago. The connection between certain quantum systems and the RH is even well-known enough to be discussed at length in some of the recent popular books on the Reimann Hypothesis. So supposing this proof were correct, then it has already been demonstrated by numerous other people.

But seeing as it's not a valid proof (which all these other people have recognized), there's not much to worry about. None of those people are going to claim you "stole their proof".

 

[eljose]

if they discovered the proof long ago...why did they not publish it?...in a post above i have proof that the potential exist and even calculated it to first order in perturbation theory depending on \delta{E(n)} and the E_{n} and E^0_{n} are known even more if we call Z the inverse of the function \zeta(s) then we could write: E_{n}=i(1/2-Z(0)) (i have oly inverted the function, and i have used simple integral equation theory to prove the existence of the potential...

The problem with Matt (as happen with most of math teachers) is that they have assumed certain conceptions in math and if you are out of these,you are nothing,i don,t know what argument will now matt grime have to say my maths are wrong,but is only an "approach to the potential" (is would be only correct to first order in perturbation theory, the whole serie of values of energy is perhaps even divergent) and the WKB is also an approach,to say that we can choose some functions that are on L^{2}(R) function space...\psi=Asen(S(x)/\hbar) for example.

another question as we are dealing with rigour an other things...can anyone of you brilliant,smart intelligent mathematician to prove the existence of infinitesimals,i,ll put even more easier,write (with numerical value) an infinitesimal...

 

[Zurtex]

Hyper real numbers have infinitesimals in them:

http://mathforum.org/dr.math/faq/analysis_hyperreals.html

http://en.wikipedia.org/wiki/Hyperreal_number

But what has that got to do with this?

Anyway, eljose, I am unsure if the mathematics in your post makes any sense or not, it could well do, but I can clearly see it does not show that all s in the critical line of \zeta (s) = 0 satisfy \Re (s) = 1/2.

And with all due respect, my 12 year old brother could see sentences in your proof that logically made no sense.

 

[eljose]

hyperreal numbers including infinitesimal..but could you write "an" infinitesimal?..

i have shown that the potential (and calculated it too) for \zeta(1/2+is) is real,(the proof is that for any existing E_{n} also E*{n}=E_{k} is also an energy from this we deduce using the expected value of the Hamiltonian that V=V* so V is real,as you can see this only happens with a=1/2 the other cases there are complex energies in the form E*{n}+(2a-1)i, but a complex energies will come from a complex potential so \zeta(a+is)=0 can not have any real root except a=1/2 that have all the roots real as the potential is real) then i have also calculated (given an integral expression for) the potential upto first order in perturbation theory.

I am checking my grammar and spelling to make it the most clearer of possible zurtex,i am not ofended .

 

[symplectic_manifold]

Oh Jesus Christ and Holy Maria!
What a thread what a thread?!
I can't spare my time for this post so I just refer to the last post on V.S.'s "proof" of Fermat.

 

[HallsofIvy]

Can I jump in just to point out that you can't prove a theorem in MATHEMATICS by using PHYSICS? The truth of a mathematics statement does not depend upon whether the physics statements are true or not.

 

[David]

if they discovered the proof long ago...why did they not publish it?

Precisely for the reasons I said above: It's not a proof! People have made the observation you think you are making even more concretely but they all acknowledge that it is not a proof of anything. So what we are saying is that even if you turn this into a coherent statement, it is something that others have already said.

...in a post above i have proof that the potential exist

You don't need to prove that there is a potential that relates to the properties of the zeta function. That has already been done. It's just that it doesn't have anything to do with a proof.

and even calculated it to first order in perturbation theory depending on \delta{E(n)} and the E_{n} and E^0_{n} are known even more if we call Z the inverse of the function \zeta(s) then we could write: E_{n}=i(1/2-Z(0)) (i have oly inverted the function, and i have used simple integral equation theory to prove the existence of the potential...

As soon as you say you have calculated something to first order, or used the WKB approximation, or anything like that we immediately KNOW that those statements can't be part of a proof.

People can already calculate all the zeros of the zeta function. But it will take an infinite amount of time, just as your approximate processes might be able to do the same calculation if you do it properly, but it still has nothing to do with a proof because an infinite process will never be considered a proof seeing as you can't complete it.

The problem with Matt (as happen with most of math teachers) is that they have assumed certain conceptions in math and if you are out of these,you are nothing,

Why exactly are you interested in trying to show anything to mathematicians if you reject mathematics? Mathematics defines its own processes very carefully so you can either accept those processes and do mathematics, or you can reject those processes and accept that you are not doing mathematics. (In that case, though, why should mathematicians listen to you when you incorrectly claim to be doing mathematics?)

i don,t know what argument will now matt grime have to say my maths are wrong,but is only an "approach to the potential" (is would be only correct to first order in perturbation theory, the whole serie of values of energy is perhaps even divergent) and the WKB is also an approach,to say that we can choose some functions that are on L^{2}(R) function space...\psi=Asen(S(x)/\hbar) for example.

another question as we are dealing with rigour an other things...can anyone of you brilliant,smart intelligent mathematician to prove the existence of infinitesimals,i,ll put even more easier,write (with numerical value) an infinitesimal...

I addressed the problem of using approximations above. As for the issue of infinitesimals, which is completely off-topic for this discussion, infinitesimals are not something whose existence should be proven - they are something that is defined. And if you knew that you would also know that it is impossible to write a numerical value for one because, by definition, they do not admit numerical expansions.

People have been pretty patient with you and tried to help you understand why what you are doing is not proof but you seem to refuse to even learn what "proof" means in mathematics and, instead, complain that mathematicians won't listen to you. Why should anybody listen to you tell them how to do their work when you have demonstrated that you don't understand what their work is?

Given that people have been trying to help you, don't you think it is pretty rude of you to refuse to listen or learn?

 

[eljose]

-To Hallsoft Ivy:this problem can be seen in a mathematical way as the Hilbert-Polya conjecture (a selfadjoint operator with real potential that has its eigenvalues as the roots of the function \zeta(1/2+is),that is to prove that exist a real potential and a self adjoint operator with a given V so all its eigenvalue are the roots of the function zeta evaluated at 1/2+is,then i prove this is self-adjoint (as the potential is real and a Hamiltonian with a real potentia has all its eigenvalues real) for the cases a+is with a different from 1/2 there are compplex roots so the potential can not be real...all the steps i made are justified mathematically i have not introduced any physical or empirical proof. (you can view H as a self-adjoint operator,and WKB solutions as the solutions to an equation of the form:
ey``+f(x)y=0 with e an small parameter e<<1 as you can see all is math there


-To symplectic manifold...if this proof were made by an universtiy smart and snob teacher i,m sure that you would accept it...

-To David:i have proved that the potential exist and have given an expression for it in the form:

V(x)=\int_{-\infty}^{\infty}dnR(n,x)\delta{E(n)} (1) (although of course this is only an approximation,you can calculate it in finite time by integrating upto a N finite so you only should need to calculate a finite (but big ) number of roots of Z(1/2+is)).

as you can see the potential exists and can be calculated,also for a=1/2 the potential is real i have calculated all the necessary steps to prove RH but if they don,t want to give you the fame and the prize because you are not famous they will invent any excuse...i am sure that if my steps (as i have said before) were made by a famous mathematician from a famous university the RH would have been proved long ago...as i have told before at least Gauss,Euler and others were given an opportunity to publish their ideas...my teachers don,t want to know anything from me or give me the chance to prove RH...

From the mathematical point of view my proof of RH is similar to this:

a differential operator H=aD^{2}+V(x) where we must choos V so the eigenvalues of this operator are precisely the roots of \zeta(1/2+is)=0.

for the potential V knowing some of the energies E_{n} we could obtain V in an integral form as expressed by (1).

for a small a<<1 (in our case is m>>h) the WKB solutions are the solutions of a differential equation ey´´+f(x)y=0..

i have not made any physical assumptions at all.

 

[Zurtex]

Let K be the set of all solutions that satisfy \zeta (s) = 0. Let there exist some p such that:

\frac{\partial^2 p}{\partial i^2} a^3 - \Gamma (i^2) = 0

Now I can prove that these exists a solution i = f_k (p) therefore i exists. If i exists there must be a limit to g(p) and p approaches infinity and therefore p exists. Thus the roots of:

\int_0^\infty \frac{g(s)}{p \Gamma(s)} ds = \sin i

Are synonyms to the Zeta functions roots and all have roots of \Re (p) = 1/2

Therefore RH is proven and no one can come up with a counter-example.

 

[Hurkyl]

You were warned.