Omega-theorems for the Hurwitz zeta-function (Q1112101)

Let \(0<\alpha \leq 1\), \(s=\sigma +it\) and let \(F(s)=\zeta (s,\alpha)\) denote \(\sum^{\infty}_{n=0}(n+\alpha)^{-s}\), \((\sigma >1)\), and its analytic continuations. The following sums are proved for a dense set of irrationals \(\alpha\). Theorem 1: We have \(\overline{\lim}_{t\to \infty}\{| F(+it)| Exp(-\frac{(\log t)^{1/2}}{\log \log t})\}=\infty.\) Theorem 2: Let \(<\sigma <1\). Then \(\overline{\lim}_{t\to \infty}\{| F(\sigma +it)| Exp(-\frac{(\log t)^{1-\sigma}}{(\log \log t)^{1+\sigma}})\}=\infty.\) Theorem 3: We have \(\overline{\lim}_{t\to \infty}\{| F(1+it)| Exp(-\frac{\log \log \log t}{\log \log \log \log t})\}=\infty.\) In a later paper (to appear) we have improved theorem 3 as follows: Theorem 4: For all fixed \(\alpha\) in \(0<\alpha \leq 1\) we have \[ \overline{\lim}_{t\to \infty}\{| F(1+it)| (\log \log t)^{- 1}\}\geq. \]