The Argument of the Riemann Ξ Function off the Critical Line

We examine the behaviour of the zeros of the real and imaginary parts of ξ(s) on the vertical line Rs = 1/2 + λ, for λ 6= 0. This can be rephrased in terms of studying the zeros of families of entire functions A(s) = 1 2 (ξ(s+λ)+ξ(s−λ)) and B(s) = 1 2i (ξ(s+λ)−ξ(s−λ)). We will prove some unconditional analogues of results appearing in [3], specifically that the normalized spacings of the zeros of these functions converges to a limiting distribution consisting of equal spacings of length 1, in contrast to the expected GUE distribution for the same zeros at λ = 0. We will also show that, outside of a small exceptional set, the zeros of Rξ(s) and Iξ(s) interlace on Rs = 1/2 + λ. These results will depend on showing that away from the critical line, arg ξ(s) is well behaved.