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 $(\log N)^{1/4}(\log\log N)^{1/4}$.
Partial sums of random multiplicative functions and extreme values of a model for the Riemann zeta function
Zeros of higher derivatives of Riemann zeta function
Zeroes of partial sums of the zeta-function
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