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We improve on previous upper bounds for the $q$th norm of the partial sums ofthe Riemann zeta function on the half line when $0<q\leqslant 1$. Inparticular, we show that the 1-norm is bounded above by $(\logN)^{1/4}(\log\log N)^{1/4}$.

Upper bounds for $L^q$ norms of Dirichlet polynomials with small $q$