| Abstrakt: |
It is proved that if T is sufficiently large, then uniformly for all positive integers ℓ⩽(logT)/(log2T)$\ell \leqslant (\log T) / (\log _2 T)$, we have maxT⩽t⩽2Tζ(ℓ)1+it⩾eγ·ℓℓ·(ℓ+1)−(ℓ+1)·log2T−log3T+O(1)ℓ+1,\begin{eqnarray*} &&\hspace*{13pc} \max _{T\leqslant t\leqslant 2T}{\left|\zeta ^{(\ell)}{\left(1+it\right)}\right|} \geqslant e^{\gamma }\cdot \ell ^{\ell }\cdot (\ell +1)^{ -(\ell +1)}\\ &&\hspace*{13pc}\quad \cdot {\left(\log _2 T - \log _3 T + O(1)\right)}^{\ell +1} \,, \end{eqnarray*}where γ is the Euler constant. We also establish lower bounds for maximum of |ζ(ℓ)(σ+it)|$ |\zeta ^{(\ell)}(\sigma +it) |$ when ℓ∈N$\ell \in \mathbb {N}$ and σ∈[1/2,1)$\sigma \in [1/2, \,1)$ are fixed. [ABSTRACT FROM AUTHOR] |
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