Semiclassical formula for the number variance of the Riemann zeros
Semiclassical formula for the number variance of the Riemann zeros
复制标题
黎曼零点数方差的半经典公式
DOI：
--
复制
发表时间：
1988
期刊：
影响因子：
--
通讯作者：
M. Berry
中科院分区：
文献类型：
--
作者：
M. Berry
假设黎曼零点的象散部分Em是量子哈密顿量的本征值，其对应的经典轨迹是混沌的，没有时间反演对称性，通过渐近论证可以得到在期望数为L的区间内，第x个零点附近的实际零点数与平均零点数之间的均方差V（L;x）的公式。这预示着当L >Lmax时，V将围绕平均值pi-2（lnln（E/2 pi）+1.4009）有准随机振荡。与V（L;由Odlyzko（1987）从105个零点Em在x=1012附近计算出的Em（x）证实了半经典预言的所有细节都在图形精度的范围内。
By pretending that the imaginery parts Em of the Riemann zeros are eigenvalues of a quantum Hamiltonian whose corresponding classical trajectories are chaotic and without time-reversal symmetry, it is possible to obtain by asymptotic arguments a formula for the mean square difference V(L;x) between the actual and average number of zeros near the xth zero in an interval where the expected number is L. This predicts that when L >Lmax, V will have quasirandom oscillations about the mean value pi -2(lnln(E/2 pi )+1.4009). Comparisons with V(L;x) computed by Odlyzko (1987) from 105 zeros Em near x=1012 confirm all details of the semiclassical predictions to within the limits of graphical precision.