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A lower bound on high moments of character sums
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abstract
For any real $k\geq 2$ and large prime $q$, we prove a lower bound on the $2k$-th moment of the Dirichlet character sum \begin{equation*} \frac{1}{\phi(q)} \sum_{\substack{\chi \text{ mod }q\\ \chi\neq \chi_0}} \Big| \sum_{n\leq x} \chi(n)\Big|^{2k}, \end{equation*} where $1\leq x\leq q$, and $\chi$ is summed over the set of non-trivial Dirichlet characters mod $q$. Our bound is known to be optimal up to a constant factor under the Generalised Riemann Hypothesis. We also get a sharp lower bound on moments of theta functions using the same method.
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Lower bounds for high moments of zeta sums
For every k>2, unconditionally, the average of |∑_{n≤x} n^{-it}|^{2k} over t∈[0,T] is ≫_k x^k (log L)^{(k-1)^2}, where L = min{x, T/x}.
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