The argument of the Riemann $Ξ$-function off the critical line
| dc.creator | Li, Xiannan | |
| dc.date | 2009-04-07 | |
| dc.date.accessioned | 2026-07-07T13:01:13Z | |
| dc.date.available | 2026-07-07T13:01:13Z | |
| dc.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. | |
| dc.description | 9 pages | |
| dc.identifier | https://arxiv.org/abs/0904.1051 | |
| dc.identifier | http://arxiv.org/abs/0904.1051 | |
| dc.identifier.uri | http://salesiana.dossiersoluciones.com/handle/123456789/226084 | |
| dc.subject | Number Theory | |
| dc.subject | 11M06; 11M26 | |
| dc.title | The argument of the Riemann $Ξ$-function off the critical line | |
| dc.type | text |