On the Berry conjecture
If \(N(T)\) is the usual counting function for zeros of \(\zeta(s)\), and \(B(x)\) is the inverse function for \({t\over 2\pi}\left(\log{t\over 2\pi}-1\right)+{7\over 8}\), the number variance is defined for \(\alpha\neq 0\) by \[ V(\alpha,T)= T^{-1}\int^T_0\Biggl\{N\Biggl(B\Biggl(x+ {\alpha\over 2}\Biggr)\Biggr)- N\Biggl(B\Biggl(x-{\alpha\over 2}\Biggr)\Biggr)- \alpha\Biggr\}^2 dx. \] \textit{M. V. Berry} [Nonlinearity 1, 399-407 (1988; Zbl 0664.10022)] has given explicit functions \(f(\alpha)\) (the `GUE part') and \(g(\alpha,T,\lambda)\) (the `arithmetic part') and conjectured that \(\widetilde V(\alpha,T,\lambda)=f(\alpha)+g(\alpha,T,\lambda)\), for some value in the range \(1\ll\lambda\ll\log T\), whenever \(\alpha\ll T\log T\). This is a much longer range for \(\alpha\) than has been previously studied. The author gives a thorough investigation of the number variance and its conjectural approximations. In particular he shows that the `simplified number variance' \[ T^{-1}\int^T_0\Biggl\{S\Biggl(t+ {2\pi\alpha \over\log(T/2\pi)}\Biggr)- S(t)\Biggr\}^2 dt \] produces the same main terms as does \(f(\alpha)+g(\alpha,t,\lambda)\), if one assumes the Riemann Hypothesis and Montgomery's pair correlation conjecture. Many other useful results are given.
scientific article; zbMATH DE number 1140495