The argument of the Riemann $Ξ$-function off the critical line
Abstract
Description
We examine the behaviour of the zeros of the real and imaginary parts of $ξ(s)$ on the vertical line $\Re s = 1/2+λ$, for $λ\neq 0$. 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 $B(s) = \frac{1}{2i} (ξ(s+λ) - ξ(s - λ))$. 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 $λ= 0$. We will also show that, outside of a small exceptional set, the zeros of $\Re ξ(s)$ and $\Im ξ(s)$ interlace on $\Re s = 1/2+λ$. These results will depend on showing that away from the critical line, $\arg ξ(s)$ is well behaved.
9 pages
9 pages