Mathematical Sciences: A Conjecture Which Implies the Riemann Hypothesis

Work to be done on this project will focus on the solution of the celebrated Riemann hypothesis. The conjecture concerns the location of the roots of a certain function of a complex variable defined initially when the real part of the variable exceeds unity. Its name is the zeta-function or Riemann zeta-function, although it was known to Leonard Euler, appearing in 1737. It is a function of fundamental importance in number theory and in the higher theory of special functions. By means of analytic continuation the zeta function can be shown to exist and be differentiable for all complex values, save for unity. Further analysis shows that all roots of the zeta-function lie in a strip in which the real part of the variable lies between zero and one. The object of this work will be to show that the roots lie along the line where the real part of the variable equals one-half. It has been known since 1914 that infinitely many roots lie along this line. The present work plans to exploit new mathematical techniques in eventually resolving the question. Specifically, the theory of Hilbert spaces whose elements are entire functions subject to certain analytic restrictions will be the central structure for the analysis. While not all such spaces are particularly useful in this context, certain of them can be shown to have elements whose roots lie along a line. The obstruction which must be cleared is that of the zeta- function's singularity. If an analogous theory can be developed allowing for isolated singularities, the result will follow. The first step in this process will be to prove the conjecture for a related class of functions called character zeta-functions. The way appears open to achieving this during the course of this award.

Work to be done on this project will focus on the solution of the celebrated Riemann hypothesis. The conjecture concerns the location of the roots of a certain function of a complex variable defined initially when the real part of the variable exceeds unity. Its name is the zeta-function or Riemann zeta-function, although it was known to Leonard Euler, appearing in 1737. It is a function of fundamental importance in number theory and in the higher theory of special functions. By means of analytic continuation the zeta function can be shown to exist and be differentiable for all complex values, save for unity. Further analysis shows that all roots of the zeta-function lie in a strip in which the real part of the variable lies between zero and one. The object of this work will be to show that the roots lie along the line where the real part of the variable equals one-half. It has been known since 1914 that infinitely many roots lie along this line. The present work plans to exploit new mathematical techniques in eventually resolving the question. Specifically, the theory of Hilbert spaces whose elements are entire functions subject to certain analytic restrictions will be the central structure for the analysis. While not all such spaces are particularly useful in this context, certain of them can be shown to have elements whose roots lie along a line. The obstruction which must be cleared is that of the zeta- function's singularity. If an analogous theory can be developed allowing for isolated singularities, the result will follow. The first step in this process will be to prove the conjecture for a related class of functions called character zeta-functions. The way appears open to achieving this during the course of this award.