Entire functions and potentials: Zeros of the Zeta function as redundant poles

Abstract

We prove the existence of a three dimensional potential, U R ( r), whose S-wave scattering amplitude has the complex zeros of the Riemann Zeta function as “redundant poles.” The potential is a C ∞ function on 0 ≤ r < ∞, and ∫ 0 ∞ r| U R ( r)| exp( αr) dr < ∞, for 0 < α < 2 λ 1, where λ 1 is the ordinate of the first zero of the ζ-function. Detailed properties of U R ( r) are given, and it is shown that its Fourier transform, U R(τ) , is meromorphic in τ, and the poles and residues are determined. The method is generalized to include a large class of even entire functions. The resulting potentials lead to scattering amplitudes with the zeros of the entire function as redundant poles.