The argument of the Riemann $\Xi$-function off the critical line

Abstract

We examine the behaviour of the zeros of the real and imaginary parts of $\xi(s)$ on the vertical line $\Re s = 1/2+\lambda$, for $\lambda \neq 0$. This can be rephrased in terms of studying the zeros of families of entire functions $A(s) = {1/2} (\xi(s+\lambda) + \xi(s - \lambda))$ and $B(s) = \frac{1}{2i} (\xi(s+\lambda) - \xi(s - \lambda))$. We will prove some unconditional analogues of results appearing in \cite{Lag}, 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 $\lambda = 0$. We will also show that, outside of a small exceptional set, the zeros of $\Re \xi(s)$ and $\Im \xi(s)$ interlace on $\Re s = 1/2+\lambda$. These results will depend on showing that away from the critical line, $\arg \xi(s)$ is well behaved.