Mathematical Sciences: Riemann Spaces of Entire Functions

德布兰日9303367 这个项目继续对复分析中的问题进行数学研究，并应用整函数的因式分解方法来解决黎曼假设。 这个古老的猜想涉及所谓的黎曼ζ函数的根。 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）。 它被认为有根沿着线垂直于真实的轴通过1/2。 经常被遗忘的是，这个函数也有负偶数的根。 这里所采用的分解理论来自自伴二阶微分算子的谱理论。 希尔伯特自己也承认，谱理论包含了黎曼假设的一个证明。 这个理论本身并不包含足够的信息根沿着临界线。 为了手头的目的，我们将致力于发展另一种谱理论。 因式分解理论实际上取代了经典的欧拉积表示的zeta函数与2 × 2矩阵，其条目是整个功能。 从矩阵的行列式中恢复了欧拉积的倒数的基本因子。 寻找黎曼zeta函数的根源问题的答案有着丰富的历史，在这一历史中，基本的数学思想和结构不断发展，并获得了自己的生命。 这个项目，无论它是否完成搜索，无疑将继续在这一传统，即研究的产品可能被证明是更有价值的目标本身。 ***

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. ***