Inverse Scattering, the Coupling Constant Spectrum, and the Riemann Hypothesis

2001/11/07 by N. N. Khuri
Physics and Astronomy · Mathematics · #hep-th #math-ph #math.AP #math.MP #math.NT

paper · pdf

published as Math.Phys.Anal.Geom. 5 (2002) 1-63 · 107 pages, 1 figure, Latex

arxiv created 2001/11/07 · arxiv updated 2009/11/30

Abstract

We use inverse scattering methods, generalized for a specific class of complex potentials, to construct a one parameter family of complex potentials V(s, r) which have the property that the zero energy s-wave Jost function, as a function of s alone, is identical to Riemann's ξ function whose zeros are the non-trivial zeros of the zeta function. These potentials have an asymptotic expansion in inverse powers of s(s-1) with real coefficients V_n(r) which are explicitly calculated. We show that the validity of the Riemann hypothesis depends essentially on simple integrability properties of the first order coefficient, V₁(r). In the case studied in this paper, this coefficient does not satisfy these conditions, but proof of that fact does indicate several possibilities for proceeding further.

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