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 $\lambda \ne 0$. 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.

9 pages