Omega theorems for zeta sums (Q1810301)

The author proves results of the following type: There exists an infinite sequence of numbers \(t_1, t_2, \ldots\) satisfying \(t_1 \geq 1\), \(t_{k+1} \geq 2 t_k\) and possessing the property that for each \(t_k\), on the interval \((1, [\sqrt{t_k/(2 \pi)}])\), there are at least \(K \geq \exp ( \frac{6}{5} \, \sqrt{ \frac{\log t_k}{\log \log t_k}}\) positive integers \(N\) such that \[ \left | \sum_{n \leq N} n^{it_k} \right |\geq \sqrt{N} \exp \left( \frac{3}{5} \, \sqrt{\frac{\log t_k}{\log \log t_k}} \right). \]