Hilbert spaces and the pair correlation of zeros of the Riemann zeta-function
Hilbert spaces and the pair correlation of zeros of the Riemann zeta-function
复制标题
希尔伯特空间和黎曼 zeta 函数的零点对相关
DOI：
10.1515/crelle-2014-0078
复制
发表时间：
2014
期刊：
影响因子：
--
通讯作者：
M. Milinovich
中科院分区：
文献类型：
--
作者：
E. Carneiro;Vorrapan Chandee;Friedrich Littmann;M. Milinovich
Montgomery的对相关猜想利用Montgomery的公式和一些指数型的极值函数，预测了函数$N(T,\beta)$的渐近行为，该函数被定义为满足$0 0$的黎曼ζ函数的非平凡零点的坐标对$\gamma$和$\gamma'$的个数。这些函数是最优的，因为它们以最小化$L^1\big(\mathbb{R}, \big\{1 - \big(\frac{\sin \pi x}{\pi x}\big)^2 \big\}\,dx\big)$ -误差的方式最大化和最小化区间$[-\beta, \beta]$的特征函数。利用整个函数的核希尔伯特空间的再现框架，给出了这一极值问题的完全解。这扩展了P. X. Gallagher在1985年的工作，在那里，$\beta \in \frac12 \mathbb{N}$的情况被认为使用了非极端的主要音和次要音。
Montgomery's pair correlation conjecture predicts the asymptotic behavior of the function $N(T,\beta)$ defined to be the number of pairs $\gamma$ and $\gamma'$ of ordinates of nontrivial zeros of the Riemann zeta-function satisfying $0 0$, using Montgomery's formula and some extremal functions of exponential type. These functions are optimal in the sense that they majorize and minorize the characteristic function of the interval $[-\beta, \beta]$ in a way to minimize the $L^1\big(\mathbb{R}, \big\{1 - \big(\frac{\sin \pi x}{\pi x}\big)^2 \big\}\,dx\big)$-error. We give a complete solution for this extremal problem using the framework of reproducing kernel Hilbert spaces of entire functions. This extends previous work by P. X. Gallagher in 1985, where the case $\beta \in \frac12 \mathbb{N}$ was considered using non-extremal majorants and minorants.