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

TL;DR

This paper investigates the zeros of the real and imaginary parts of the Riemann xi-function off the critical line, revealing that their spacings tend to be equal and differ from the GUE distribution, with zeros interlacing outside a small exceptional set.

Contribution

It provides unconditional results on the zero spacings of functions related to \xi(s) off the critical line, contrasting with the GUE distribution at \xi(s) on the line.

Findings

Zero spacings of these functions tend to equal spacings of length 1.

Zeros of  and  parts interlace outside a small exceptional set.

Normalized zero spacings differ from the GUE distribution.

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.