Mathematical Sciences: A Conjecture Which Implies the Riemann Hypothesis

在这个项目中要做的工作将集中在著名的黎曼假设的解上。这个猜想是关于一个复数变量的函数的根的位置，当变量的实部超过单位时，该函数被定义为复数变量。它的名字是ζ函数或黎曼ζ函数，尽管它是由伦纳德·欧拉在1737年发现的。它是数论和更高级的特殊函数理论中最重要的一个函数。通过解析延拓的方法，可以证明zeta函数对于除单位外的所有复值都是存在且可微的。进一步分析表明，函数的所有根都位于变量实部介于0和1之间的条上。这项工作的目的是证明根位于变量实部等于1 / 2的直线上。自1914年以来，人们就知道这条线上有无限多根。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，其元素是受某些解析限制的完整函数，将是分析的中心结构。虽然并非所有这样的空间在这种情况下都特别有用，但可以证明其中某些空间具有根位于一条线上的元素。必须清除的障碍是ζ函数的奇异性。如果一个类似的理论可以发展，允许孤立的奇点，结果将随之而来。这个过程的第一步将是证明一个相关的函数类的猜想，称为字符ζ函数。在这个奖项的过程中，实现这一目标的方式似乎是开放的。

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.