Richert proved in https://link.springer.com/article/10.1007/BF01399533 that $$ \zeta(s) =O\left( |\Im(s)|^{100(1-\Re(s))^{3/2}} (\log |\Im(s)|)^{2/3}\right)$$ uniformly in the region $\Re(s)\in [1/2,1], |\Im(s)|\geq 2.$ The exponent $100$ has been improved to $4.45$ by Ford https://arxiv.org/abs/1910.08209.
My question is whether the constant $4.45$ has been further improved?
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 $\sigma(\lambda)\leq\sigma\leq 1$.
In addition to GH from MO's answer, if one wishes to keep the $(\log|\Im(s)|)^{2/3}$ factor, then there is a very recent improvement due to Bellotti [1]. In particular, Bellotti proved that
$$ \zeta(s)\leq 70.7|\Im(s)|^{4.43795(1-\Re(s))^{3/2}}(\log|\Im(s)|)^{2/3} $$
for $|\Im(s)|\geq 3$ and $\frac{1}{2}\leq \sigma\leq 1$. That is, she obtains an exponent of $4.43795$.
[1] C. Bellotti, Explicit bounds for the Riemann zeta function and a new zero-free region, Journal of Mathematical Analysis and Applications, 2024 (or https://arxiv.org/abs/2306.10680).
