Mathematical Sciences: Riemann Spaces of Entire Functions

这个项目继续对复杂分析中的问题进行数学研究，并应用整个函数的因式分解方法来尝试解决黎曼假设。这个古老的猜想涉及到所谓黎曼ζ函数的根。这是数论中的一个重要函数，它本身不是完整的（在数字1处有一个极点）。它的根沿着垂直于穿过1/2的实轴的直线。经常被遗忘的是这个函数在负偶数上也有根。本文采用的分解理论来源于自伴随二阶微分算子的谱理论。希尔伯特自己猜想，谱理论包含了黎曼假设的证明。这个理论本身并没有包含足够的关于沿临界线的根的信息。为了手头的目的，我们将努力发展另一种光谱理论。因式分解理论实际上用二乘二矩阵取代了经典的欧拉积表示，其元素是整个函数。欧拉积的倒数的初等因子是从矩阵的行列式中得到的。寻找黎曼ζ函数的根问题的答案有着丰富的历史，在此过程中，基本的数学思想和结构得到了发展，并获得了自己的生命。这个项目，无论它是否完成搜索，毫无疑问将继续这一传统，即研究的产品可能被证明比目标本身更有价值。＊＊＊

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. ***