" THE IMPORTANCE OF CHEBYSHEV FUNCTION IN RIEMANN HYPOTHESIS"
The Chebyshev function , giving the sum of f(x)=log(x) (natural logarithm) over primes and prime powers p,p^2 , p^3 , p^4,..... has been widely studied V.Mangoldt himself using the 'Argument principle' for Complex integrals managed to prove that this function Psi(x) could be written as a sum over all the Non-trivial zeros of the Riemann zeta function.
Form Mangoldt's result Hadamard showed that there couldn't be Non-trivial roots with real part 1 or 0, hence the Prime Number theorem was equivalent to Psi(x)/x ---->1 as x-->oo.
Recently an even more feature of this Chebyshev function, involves the existence of a HIlbert-Polya operator H so Z(1/2+iH)|n>=0 with H|n>=E_n |n>, recent investigations
http://www.wbabin.net/science/moreta8.pdf
Have lead to a proof that no matter what the operator H is , its trace Tr{exp(iuH)} (u > 0 and real number ) is related to the derivative of Psi(x) in terms of theory of distributions
evaluated at x=exp(u), this is a general result that would yield to a proof of RH, the author of the .pdf file given above describes a curious manner using Semi-classical approach of Hamiltonian Quantum mechanics so:
Tr{exp(iuH)} = Int(-oo,oo)dxexp(iuV(x)) deducing from this Non-linear integral equation a
form for the inverse of potential V(x) , f(x) so (V o f) (x)=x.
The author solves the integral equation using a Fourier integral (understood in Principal Value) involving Tr{exp(iuH)} , paper has been published at 'General Science JOurnal of Mathematical Physics' , the article has been reviewed as a hope to derive the long- standing Riemann Hypothesis proposed by Riemann in 1859 from the existence of a Hamiltonian having its 'Energy levels' equal to the complex part of the Riemann zeta function Non-trivial zeros.
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