Mathematical Sciences: Riemann Spaces of Entire Functions

De Brange 9303367这个项目继续对复数分析中的问题进行数学研究，并应用整函数因式分解的方法来尝试解决黎曼假设。这个古老的猜想涉及到所谓的Riemann Zeta函数的根源。这是数论中的一个重要函数，它本身并不完整(在数字1处有一个极点)。它被认为是沿着与实轴垂直通过1/2的直线有根的。人们经常忘记的是，这个函数也有根在负的偶数。本文所用的因式分解理论源于自伴二阶微分算子的谱理论。希尔伯特自己猜测，光谱理论包含了黎曼假设的证明。这一理论本身并不包含关于临界线上的根的足够信息。为了手头上的目的，我们将努力开发另一种光谱理论。因式分解理论实际上用2乘2矩阵代替了zeta函数的经典欧拉积表示，矩阵的条目是整函数。从矩阵的行列式中求出欧拉积的倒数的基本因子。寻找Riemann Zeta函数根问题的答案有着丰富的历史，在这个过程中，基本的数学思想和结构得到了发展，并获得了自己的生活。这个项目，无论它是否完成了搜索，无疑都将延续这一传统，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. ***