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What is a standard reference for the simple fact that, for $T$ fixed and $\sigma\to -\infty$, every derivative $|(1/\zeta)^{(r)}(\sigma+i T)|$ of the Riemann zeta function decreases faster than any exponential on $\sigma$?

Here is a quick sketch of a proof: use the functional equation $\zeta(s) = \zeta(1-s) 2^s \pi^{s-1} \Gamma(1-s) \sin \frac{\pi s}{2}$, together with the fact that any derivative $(1/\Gamma)^{(j)}(\sigma+iT)$ decreases faster than any exponential for $T$ fixed and $\sigma\to \infty$. In turn, this fact about $(1/\Gamma)^{(j)}(\sigma+iT)$ follows easily from the following: for $T$ fixed, $\Gamma(\sigma+iT)$ grows faster than any exponential as $\sigma\to \infty$ (by Stirling's formula), whereas, again for $T$ fixed and $\sigma\to \infty$, $(\Gamma'/\Gamma)(\sigma+i T)$ is bounded by $O(\log \sigma)$ and every derivative of $(\Gamma'/\Gamma)$ is $O(1)$ (see, e.g., appendix C.1 in Montgomery-Vaughan).

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    $\begingroup$ You can reduce the statement to $r=0$ by applying Cauchy's formula for derivatives. Of course you need to assume that $T\neq 0$ otherwise the trivial zeros of $\zeta(s)$ invalidate the statement. $\endgroup$
    – GH from MO
    Commented Dec 17, 2023 at 11:26
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    $\begingroup$ ,Ah, and then it's just the functional equation, together with the fact that $\Gamma(\sigma+iT)$ grows taster than any exponential as $\sigma\to \infty$. $\endgroup$
    – H A Helfgott
    Commented Dec 17, 2023 at 11:42

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