A hybrid Euler-Hadamard product for the Riemann zeta function
A hybrid Euler-Hadamard product for the Riemann zeta function
复制标题
黎曼 zeta 函数的混合 Euler-Hadamard 积
DOI：
10.1215/s0012-7094-07-13634-2
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发表时间：
2007
影响因子：
2.5
通讯作者：
J. Keating
中科院分区：
文献类型：
--
作者：
S. Gonek;C. Hughes;J. Keating
我们使用一个平滑版本的显式公式找到一个准确的逐点逼近的黎曼zeta函数作为一个产品在其非平凡的零乘以产品在素数。我们用随机矩阵的特征多项式对第一乘积进行建模。这提供了一个zeta函数的统计模型，它以一种自然的方式涉及素数。然后，我们采用该模型在一个启发式计算的时刻的zeta函数的模的临界线。对于第二个和第四个时刻，我们严格地建立了我们方法中的所有步骤。这一计算阐明了这些时刻的基础上与随机矩阵理论的连接最近progratures
We use a smoothed version of the explicit formula to find an accurate pointwise approximation to the Riemann zeta function as a product over its nontrivial zeros multiplied by a product over the primes. We model the first product by characteristic polynomials of random matrices. This provides a statistical model of the zeta function which involves the primes in a natural way. We then employ the model in a heuristic calculation of the moments of the modulus of the zeta function on the critical line. For the second and fourth moments, we establish all of the steps in our approach rigorously. This calculation illuminates recent conjectures for these moments based on connections with random matrix theory