Mathematical Sciences: A Conjecture Which Implies the Riemann Hypothesis

在这个项目上要做的工作将集中在著名的黎曼假说的解决上。这个猜想涉及复变量的某个函数的根的位置，当变量的实部超过1时，最初定义该函数的根。它的名字是Zeta-Function或Riemann Zeta-Function，尽管伦纳德·欧拉知道它是在1737年出现的。在数论和高等特殊函数理论中，它是一个非常重要的函数。通过解析延拓的方法，可以证明Zeta函数是存在的，并且对所有复值都是可微的，除了单位。进一步的分析表明，zeta函数的所有根都位于变量的实部介于0和1之间的带状区域中。这项工作的目的是证明根位于变量实部等于一半的直线上。自1914年以来，人们就知道，沿着这条线有无限多的根。目前的工作计划利用新的数学技术来最终解决这个问题。具体地说，希尔伯特空间的理论将是分析的中心结构，其元素是受一定分析限制的整函数。虽然不是所有这样的空间在这种情况下都特别有用，但可以显示其中某些空间具有根沿一条线的元素。必须清除的障碍是Zeta函数奇点的障碍。如果能发展出一种允许孤立奇点的类似理论，结果就会随之而来。这一过程的第一步将是证明一类相关函数的猜想，这类函数称为字符Zeta函数。在这个奖项的过程中，实现这一目标的道路似乎是敞开的。

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.