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...