Mathematical Sciences: A Conjecture Which Implies the Riemann Hypothesis

该项目要做的工作将集中于解决著名的黎曼假设。 该猜想涉及当变量的实部超过单位时最初定义的复变量的某个函数的根的位置。 它的名字是 zeta 函数或黎曼 zeta 函数，尽管 Leonard Euler 在 1737 年就知道了它。它是数论和特殊函数的高级理论中具有根本重要性的函数。 通过解析延拓，zeta 函数可以被证明存在并且对于所有复值都是可微的，除了单位。 进一步分析表明，zeta 函数的所有根都位于一条带中，其中变量的实部位于 0 和 1 之间。 这项工作的目的是证明根位于变量实部等于二分之一的线上。 自 1914 年以来，人们就知道这条线上有无数的根源。 目前的工作计划利用新的数学技术最终解决这个问题。 具体来说，希尔伯特空间理论将成为分析的中心结构，其元素是受某些分析限制的完整函数。 虽然并非所有此类空间在这种情况下都特别有用，但其中某些空间可以显示其根沿线排列的元素。 必须清除的障碍是 zeta 函数的奇异性。 如果可以发展出一种允许孤立奇点的类似理论，结果就会随之而来。 此过程的第一步是证明有关称为字符 zeta 函数的相关函数类的猜想。 在这个奖项的评选过程中，实现这一目标的道路似乎是开放的。

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.