Inverse scattering, the coupling constant spectrum, and the Riemann hypothesis

Abstract

It is well known that the s-wave Jost function for a potential, λV, is an entire function of λ with an infinite number of zeros extending to infinity. For a repulsive V, and at zero energy, these zeros of the ‘coupling constant’, λ, will all be real and negative, λn(0)<0. By rescaling λ, such that λn<−1/4, and changing variables to s, with λ=s(s−1), it follows that as a function of s the Jost function has only zeros on the line sn=1/2+iγn. Thus, finding a repulsive V whose coupling constant spectrum coincides with the Riemann zeros will establish the Riemann hypothesis, but this will be a very difficult and unguided search.