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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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}.