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

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