Heath-Brown (2016) proved that $$\zeta(\sigma+it)\ll_\varepsilon t^{\frac{1}{2}(1-\sigma)^{3/2}+\varepsilon},\qquad 0\leq\sigma\leq 1,\qquad t\geq 1.$$ The exponent $1/2$ can be improved to $\frac{8}{63}\sqrt{15}=0.4918\dots$ for $1/2\leq\sigma\leq 1$. Moreover, for any $\lambda>\frac{2}{\sqrt{27}}=0.3849\dots$, there exists $\sigma(\lambda)<1$ such the exponent $1/2$ can be improved to $\lambda$ for $\sigma(\lambda)\leq\sigma\leq 1$.

Heath-Brown (2016) proved that, for any $\varepsilon>0$, $$\zeta(\sigma+it)\ll_\varepsilon t^{\frac{1}{2}(1-\sigma)^{3/2}+\varepsilon},\qquad 0\leq\sigma\leq 1,\qquad t\geq 1.$$ The exponent $1/2$ can be improved to $\frac{8}{63}\sqrt{15}=0.4918\dots$ for $1/2\leq\sigma\leq 1$. Moreover, for any $\lambda>\frac{2}{\sqrt{27}}=0.3849\dots$, there exists $\sigma(\lambda)<1$ such the exponent $1/2$ can be improved to $\lambda$ for $\sigma(\lambda)\leq\sigma\leq 1$.