Explicit bound on $\zeta(s)$ inside a zero-free region?

Question

Does anybody know of a place in the literature where one can find an explicit result of the form $|\zeta(\sigma+it)|\leq C \log t$ for $t$ within a zero-free region (assuming $t$ is larger than an explicit constant)?

There is Theorem 6.25 in

https://personalpages.manchester.ac.uk/staff/mark.coleman/old/MATH41022/Present/Notes/Notes%206d%20PNT%202018-19.pdf

-- that is exactly the sort of result I'm looking for, but I'd prefer to refer to something that has been actually published (or at least posted on the arxiv).

This kind of result is not at all hard, but it would make no sense to prove it from scratch (given that it must have been proved many times before).

Answer 1

Mark Coleman just answered: the result in his notes is Prop. 5.3.2 in G. J. O. Jameson's The Prime Number Theorem.

Answer 2

Kevin Ford proved that if $|t|\geq 3$ and $\frac{1}{2}\leq\sigma\leq 1$, then $$ |\zeta(\sigma+it)|\leq 76.2|t|^{4.45(1-\sigma)^{3/2}}(\log|t|)^{2/3}. $$ This leads to his highly cited explicit version of the Vinogradov--Korobov zero-free region for $\zeta(\sigma+it)$. For $\sigma$ in any known zero-free region for $\zeta(\sigma+it)$, the bound is essentially of size $(\log|t|)^{2/3}$ when $|t|$ is large. Since RH has been verified in the range $|t|\leq 3.0001\times 10^{12}$, in which case one can do much better than a bound of the form $C\log|t|$, Kevin's result is strictly better than $\frac{1}{3}\log|t|$ for $|t|>3\times 10^{12}$, even if you venture all the way to the edge of Kevin's (so far best known) explicit version of the Vinogradov--Korobov zero-free region.