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The following is a well-known convexity bound for Dirichlet $L$-functions.

Theorem Let $\chi$ be a primitive character to the modulus $q$. Then, for any fixed $\varepsilon>0$ and any $k\in\mathbb{Z}_{\ge 0}$ we have \begin{equation*} L^{(k)}(\sigma+it,\chi)\ll\begin{cases} (qt)^{1/2-\sigma+\varepsilon}&\text{if } \sigma\le 0\\ (qt)^{1/2(1-\sigma)+\varepsilon}&\text{if }0\le\sigma\le 1\\ (qt)^{\varepsilon}&\text{if }\sigma\ge 1 \end{cases} \end{equation*}

I want to know if we can make this explicit in terms of writing the $\varepsilon$ more explicitly. That is, can we write the $\varepsilon$ term as powers of logarithms?

I expect that one might use Cauchy's estimate for derivatives to repeatedly differentiate to obtain bounds for higher derivatives, and doing this morally introduces a new logarithmic term, so it would be nice if we could make as explicit as possible the above convexity bound.

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Usually, one just forgoes the $\epsilon$ in terms of a completely explicit result. That is, the implicit constant in the $\ll$ symbol is computed and the exponent $\delta+\epsilon$ for $\delta \le \frac{1}{4}$ is replaced with an explicit positive number say $\delta'$. These results belong to the relm of "explicit number theory" or "explicit (sub)convexity estimates". For some references in the case of $ k = 1$:

https://arxiv.org/pdf/2206.11112 - This 2022 paper contains an explicit subconvexity estimate of Burgess strength for prime power modulus

https://arxiv.org/pdf/2302.13444 - This 2023 paper gives a better result in the case of the Riemann zeta function

https://arxiv.org/pdf/2208.11123 - A 2022 result for primitive characters only

A collection of these results (and some older ones) can be found in https://arxiv.org/pdf/2410.06082.

For $k > 1$, you can use the above results and apply Cauchy's integral formula as you say.

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