Thread: Is V. Mangoldt formula just a trace?...

Extracted from the webpage Chebyshev function - Wikipedia, the free encyclopedia

If we differentiate the expression inside V. Mangoldt formula we get:

1-df-(x^3 -x)^{-1}= Sum(r)x^{r-1} but if RH is true then r (Non trival zeros) are of the form r=1/2+iE(n) , multiplying both sides by x^{0.5} and putting x=exp(u) we have just a "Partition function" which is just the trace of a certain operator

Z=Tr{e^{iuH}) where f= Chebyshev function d=differential operator.

A better explanation is inside this webpage....or in the paper: (arxiv)

http://arxiv.org/ftp/math/papers/0607/0607095.pdf

Where the author relates Statistical Mechanics and Number Theory.... although for me is a bit confusing.

I have no idea what you are saying but the only thing that I am first concered about whether or not you can differenticate. Is this a differenciable function? Remember these funtions are from number theory they might be discrete and hence non differenciable.

Originally Posted by lokofer

Extracted from the webpage Chebyshev function - Wikipedia, the free encyclopedia

If we differentiate the expression inside V. Mangoldt formula we get:

1-df-(x^3 -x)^{-1}= Sum(r)x^{r-1} but if RH is true then r (Non trival zeros) are of the form r=1/2+iE(n) , multiplying both sides by x^{0.5} and putting x=exp(u) we have just a "Partition function" which is just the trace of a certain operator

Z=Tr{e^{iuH}) where f= Chebyshev function d=differential operator.

A better explanation is inside this webpage....or in the paper: (arxiv)

http://arxiv.org/ftp/math/papers/0607/0607095.pdf

Where the author relates Statistical Mechanics and Number Theory.... although for me is a bit confusing.

Link is dead.

RonL

sorry the link was:

http://arxiv.org/PS_cache/math/pdf/0607/0607095.pdf

the "pdf" link is out i think this is only a postscript... a earlier .pdf version (incomplete) is avaliable...

or check the whole abstract... Format selector for math/0607095 (choose your favourite format).

I've looked it over (briefly). I can't say I understand most of it, but at the same time neither did I give it an appropriate amount of time. I'm sure they had some specific problem in mind when they did this, but the term "frigging mess" comes to mind!

A quick rundown from my perspective (which, I'm sorry to say, doesn't address the original question) is that the author is trying to express the Hamiltonian operator in Mechanics in such a way that the energy eigenvalues take on a particular kind of format. I have seen such methods before and they tend to be specific to the form of the potential function. This one may be more general, I'm not quite sure.

Unfortunately the Mangoldt formula is where I lose track of what they are doing. It appears the author is using it to, uh, "simplify" the appearance of the partition function so the eigenvalues of the Hamiltonian can be extracted more easily. Number theory enters because the author is assuming a discrete, rather than continuous, spectrum for the eigenvalues.

I'll admit I'm intrigued by the use of the Chebyshev polynomials in this kind of problem. I wouldn't have thought they would have any application unless V was of a (uselessly specific) particular form.

-Dan

As far i understood...(Post script isn't very clear) author is trying this (¿?)

- Find a Hamiltonian H=p^2 +V(x) (the most usual Hamiltonian in one dimension) so the energies are just the imaginary part of the Non-trivial zeros.

- Using Mangoldt formula and differnentiating he gets an expression involving Sum(n)exp(iuE(n)) ...(for pure complex u=-is this is just a partition function of statistical mechanics) then he approximates the sum over eigenvalues by an integral involving H (approximation) which he solves getting the inverse function of the potential V(x) ,which he calls f(t) given by an inverse Fourier transform... i'll try to contact him to ask for the paper.

- The approximation Sum(n)exp(iuE(n))=Int(-oo,oo)dxdpexp(iuH) is very well known in physics and called "Semiclassical (or WKB) approach"... I'm a physicist so i can't understand some details of the paper...

Originally Posted by lokofer

As far i understood...(Post script isn't very clear) author is trying this (¿?)

- Find a Hamiltonian H=p^2 +V(x) (the most usual Hamiltonian in one dimension) so the energies are just the imaginary part of the Non-trivial zeros.

- Using Mangoldt formula and differnentiating he gets an expression involving Sum(n)exp(iuE(n)) ...(for pure complex u=-is this is just a partition function of statistical mechanics) then he approximates the sum over eigenvalues by an integral involving H (approximation) which he solves getting the inverse function of the potential V(x) ,which he calls f(t) given by an inverse Fourier transform... i'll try to contact him to ask for the paper.

- The approximation Sum(n)exp(iuE(n))=Int(-oo,oo)dxdpexp(iuH) is very well known in physics and called "Semiclassical (or WKB) approach"... I'm a physicist so i can't understand some details of the paper...

I, too, am a Physicist and I didn't catch the WKB approximation part. Ah, well, semi-classical Physics isn't my forte anyway!

-Dan

I think he's referring to the fact that you can get the "Partition function" in 2 ways:

- Quantum:-->(exact) sum over all eigenvalues Sum(n) exp(iuE(n))

-Semi-classical:--> Int(-oo,oo)dxdpexp(iuH) H=p^{2}+V(x) (here x an p are just "continous" variables)

He approaches the sum by an integral...(at first order approximation, for more accurate one he could have applied Euler-Mc Laurin formula involving the derivatives and so on...., but i myself can't indicate how since i know the formula only for a variable "x" )...