A variational approach to the explicit formula
A variational approach to the explicit formula
复制标题
显式公式的变分方法
DOI：
10.1002/cpa.10089
复制
发表时间：
2003
影响因子：
3
通讯作者：
E. Bombieri
中科院分区：
文献类型：
--
作者：
E. Bombieri
在1859年，黎曼发现了一个精确的解析公式，可以计算素数的个数，其重要性不容低估。虽然近似公式的黎曼类型的加权计数的素数形式的基本工具的分析数论，更精细的信息所涉及的一个确切的公式很少发现直接应用于算术问题。另一方面，这些精确公式所涉及的理论方面似乎对理解与素数有关的基本问题至关重要，例如尚未解决的黎曼假设。第一个研究一个显式公式的一大类测试功能是由A。P. Guinand [3]在1942年，他认为它是一个变换公式，与泊松公式没有什么不同。1952年，A. Weil [4]提出了一个显式公式，在经典情形和所谓的函数域情形下具有相同的形式。他还指出，黎曼假设可以表述为由显式公式产生的某个厄米泛函的正性。在有限域上曲线的zeta函数的几何情况下，这个正性是代数指标定理的简单结论，即：
In 1859, Riemann found an exact analytic formula for the number of primes up to a given limit, the importance of which should not be underestimated. While approximate formulae of Riemann type for a weighted count of primes form a basic tool of analytic number theory, the finer information involved in an exact formula rarely finds direct application to arithmetical questions. On the other hand, the theoretical aspects involved in such exact formulae seem to be of paramount importance in understanding the fundamental questions attached to primes, such as the still-unsolved Riemann hypothesis. The first study of an explicit formula for a large class of test functions was done by A. P. Guinand [3] in 1942, who viewed it as a transformation formula not unlike Poisson’s formula. In 1952, A. Weil [4] put forward an explicit formula with an identical formulation both in the classical case and the so-called function field case. He also pointed out that the Riemann hypothesis can be stated as the positivity of a certain Hermitian functional arising from the explicit formula. In the geometric case of zeta functions of curves over finite fields, this positivity is an easy consequence of the algebraic index theorem, namely: