de Branges 9303367 This project continues mathematical research on problems in complex analysis and the application of methods of factorization of entire functions to attempt a resolution of the Riemann Hypothesis. This venerable conjecture concerns the roots of the so-called Riemann zeta-function. This an important function in number theory which itself is not entire (having a pole at the number 1). It is believed to have roots along the line perpendicular to real axis passing through 1/2. Often forgotten is the fact that this function also has roots at the negative even integers. The factorization theory employed here derives from the spectral theory of self-adjoint second order differential operators. Hilbert himself conjectured that the spectral theory contains a proof of the Riemann hypothesis. This theory in itself does not contain enough information about roots along the critical line. Work will be done in developing another spectral theory for the purposes at hand. Factorization theory in effect replaces the classical Euler product representation of the zeta-function with two-by-two matrices whose entries are entire functions. The elementary factors of the reciprocal of the Euler product are recovered from the determinants of the matrices. The search for an answer to the question of the roots of the Riemann zeta-function has a rich history, in which fundamental mathematical ideas and structures evolved and a gained life of their own. This project, whether or not it completes the search, will undoubtedly continue in this tradition whereby the products of the research may prove to be more valuable the goal itself. ***
