Mathematical Sciences: Hypergeometric Functions, Zeta and Gamma Values in Finite Characteristic

数学科学：超几何函数、有限特征中的 Zeta 和 Gamma 值

• 批准号: 9623187
• 负责人: Dinesh Thakur
• 金额: $ 4.23万
• 依托单位: University of Arizona
• 依托单位国家: 美国
• 项目类别: Standard Grant
• 财政年份: 1996
• 资助国家: 美国
• 起止时间: 1996-08-15 至 1999-07-31
• 项目状态: 已结题
• 来源: https://www.nsf.gov/awardsearch/showAward?AWD_ID=9623187&HistoricalAwards=false
• 关键词: Mathematical; Sciences; Hypergeometric; Functions; Zeta

Thakur 9623187 The investigator will explore analogies between Number Theory and Geometry. The hypergeometric function is one of the most important special functions in mathematics and mathematical physics. Many of the functions that arise in various branches of mathematics and physics-Bessel, Legendre, and Jacobi functions, and many interesting orthogonal polynomials-are special cases of the hypergeometric function. Similarly, the q-hypergeometric functions arise in the study of mathematical physics. The investigator has introduced an analogue for function fields, and he will develop their theory and applications. A classical problem in Number Theory is to understand the nature of the values of the Riemann zeta-function. Euler showed that the values at even integers s are rational multiples of powers of pi. For odd s, the only known result is that the zeta-function is irrational at s=3. Carlitz considered an analogue of the zeta-function by taking a sum over monic polynomials with coefficients in finite fields, and established an analogue of Euler's result. In the joint work with Anderson, the PI showed that for any positiveminteger s, the values are logarithms of algebraic numbers, and are therefore transcendental. It was also shown that the analogue of Euler's result for odd s does not hold in this case. The proposer plans to generalize thismresult to general function fields. The order of vanishing of zeta functions gives important arithmetic information. The PI plans to complete new partial results that he has obtained about these orders. Also, the PI plans to workmout the relations of special values with the cyclotomic theory, in light of recent work of Anderson. The special values of the classical gamma function have also been extensively studied. The PI introduced a new gamma function for function fields. He has established special values results for its interpolations at finite primes for general function fields and for interpolati on at the infinite prime only for the rational function field. He plans to settle the remaining case. This research falls into the general mathematical field of Number Theory. Number theory has its historical roots in the study of the whole numbers, addressing such questions as those dealing with the divisibility of one whole number by another. It is among the oldest branches of mathematics and was pursued for many centuries for purely aesthetic reasons. However, within the last half century it has become an indispensable tool in diverse applications in areas such as data transmission and processing, and communication systems.

Thakur 9623187 研究者将探索数论和几何之间的类比。 超几何函数是数学和数学物理中最重要的特殊函数之一。 数学和物理学各个分支中出现的许多函数——贝塞尔函数、勒让德函数和雅可比函数，以及许多有趣的正交多项式——都是超几何函数的特例。 类似地，q-超几何函数出现在数学物理的研究中。 研究人员引入了函数域的类似物，他将发展它们的理论和应用。 数论中的一个经典问题是理解黎曼 zeta 函数值的本质。 欧拉表明偶数 s 的值是 pi 幂的有理倍数。 对于奇数 s，唯一已知的结果是 zeta 函数在 s=3 时是无理数。 卡利茨通过对具有有限域系数的一元多项式求和来考虑 zeta 函数的类似物，并建立了欧拉结果的类似物。 在与安德森的合作中，PI 表明，对于任何正整数，其值都是代数数的对数，因此是超越的。 还表明，欧拉对奇数 s 的结果的类似结果在这种情况下不成立。 提议者计划将此结果推广到通用函数领域。 zeta 函数的消失顺序提供了重要的算术信息。 PI 计划完成他获得的有关这些订单的新的部分结果。此外，PI 计划根据安德森最近的工作，用分圆理论来解决特殊值的关系。 经典伽马函数的特殊值也得到了广泛的研究。 PI 为函数域引入了新的 gamma 函数。 他为一般函数域的有限素数插值和有理函数域的无限素数插值建立了特殊的值结果。 他计划解决剩余的案件。 这项研究属于数论的一般数学领域。 数论的历史根源在于对整数的研究，解决诸如一个整数能否被另一个整数整除等问题。 它是数学最古老的分支之一，几个世纪以来纯粹出于美学原因而被人们所追求。然而，在过去的半个世纪中，它已成为数据传输和处理以及通信系统等领域的各种应用中不可或缺的工具。