Zero spacing distributions for differenced L-functions

Abstract

The paper studies the local zero spacings of deformations of the Riemann ξ-function under certain averaging and differencing operations. For real h we consider the entire functions Ah(s) := 1 (ξ(s + h) + ξ(s − h)) and Bh(s) = 1 2i (ξ(s + h) − ξ(s − h)) . For |h| ≥ 1 2 the zeros of Ah(s) and Bh(s) all lie on the critical line ℜ(s) = 1 and are simple zeros. The number of zeros of these functions to height T has asymptotically the same density as the Riemann zeta zeros. For fixed |h| ≥ 1 the distribution of normalized zero spacings of these functions up to height T converge as T → ∞ to a limiting distribution, which consists of equal spacings of size 1. That is, these zeros are asymptotically regularly spaced. Assuming the Riemann hypothesis, the same properties hold for all nonzero h. In particular, these averaging and differencing operations destroy the (conjectured) GUE distribution of the zeros of the ξ-function, which should hold at h = 0. Analogous results hold for all completed Dirichlet L-functions ξχ(s) having χ a primitive character.