A Self-Referential Integral Equation for the Nontrivial Zeros of the Riemann Zeta Function

Abstract

Starting from Jensen’s integral representation of the Riemann zeta function, I derive, by a chain of convergent algebraic manipulations, a self-referential integral equation that a complex number F must satisfy in order for ρ = F ( s ) / ( F ( s ) − 1 ) to be a nontrivial zero of the zeta-function, ζ . The resulting equation, is an explicit kernel and a biconditional characterization of the F -image of the nontrivial zeros. From calculations, I show that the Riemann Hypothesis is equivalent to the statement | F | = 1 for every solution. The equation simplifies dramatically to a Dirichlet-like series of lower incomplete gamma functions evaluated on the imaginary axis. A new representation of the Jensen’s integral for the Zeta function is show to be reducible to a series relating to Von Staudt-Clausen theorem. These new relations are shown to have a direct bearing on the relationship of the non-trivial zeros of the Zeta function and the primes. I discuss the analytical constraints and outline what is needed to close the gap to a proof of the Riemann Hypothesis.