$L'/L(s,\chi)\ll\log^2 q(|t|+2)$ in the zero-free region, after removing the poles at $1$ and at the exceptional zero (Davenport §§16, 19)

The logarithmic derivative of $L(s,\chi)$ is $O(\log^2 q(|t|+2))$ in a zero-free region, once its polar parts are removed. Fix a region constant $c>0$. There is a constant $C>0$, depending only on $c$, such that for every modulus $q\ge1$, every Dirichlet character $\chi$ modulo $q$ and every exceptional set $E$ for $\chi$ with respect to $c$ (IsExceptionalSet c χ E: $E$ has at most one element; each element is a real zero $\beta\in(0,1)$ of $L(\cdot,\chi)$ lying in the region $\sigma\ge1-c/\log(q(|t|+2))$, and can exist only if $\chi$ is quadratic and non-principal; and $L(s,\chi)\ne0$ at every $s\ne1$ of that region outside $E$) the following two statements hold.

1. The exceptional zero has small multiplicity: for every $\beta\in E$, the order $m_\beta$ of the zero of $L(\cdot,\chi)$ at $\beta$ satisfies $m_\beta\le C\log(2q)$.

2. Bound in the quarter-region. For every complex $s=\sigma+it$ with

one has

where $\delta_\chi=1$ if $\chi$ is the principal character and $\delta_\chi=0$ otherwise.

In words: inside the (slightly shrunken) zero-free region, $L'/L(s,\chi)$ equals the sum of its polar parts — $-1/(s-1)$ from the pole of $L(s,\chi_0)$ at $s=1$, and $m_\beta/(s-\beta)$ from the exceptional zero — plus a remainder that is uniformly $O(\log^2 q(|t|+2))$. This is the estimate Davenport uses in §§19–20 for the integrals over the shifted contour: it comes from the local partial-fraction expansion $\frac{L'}{L}(s,\chi)=\sum_{|\rho-s_0|\le3/2}\frac{m_\rho}{s-\rho}+O(\log q(|t|+2))$ (§16), since every zero other than $\beta$ lies outside the region, so $|s-\rho|\gg1/\log q(|t|+2)$ for $s$ in the quarter-region, and there are only $O(\log q(|t|+2))$ such $\rho$. For the principal character one uses $L(s,\chi_0)=\zeta(s)\prod_{p\mid q}(1-p^{-s})$, whose finite product contributes $O(\log q)$ to the logarithmic derivative for $\sigma\ge3/4$, and the corresponding bound for $\zeta'/\zeta+1/(s-1)$ in a zero-free region of $\zeta$.

The region constant is a free parameter (as in Davenport.psi_char_of_region) so that the theorem is independent of the specific constant produced by the §14 zero-free-region theorem; for large $c$ the hypothesis may be unsatisfiable, which makes the statement vacuous there, not false. The restriction $\sigma\ge3/4$ keeps everything inside the half-plane where the growth bound $L(s,\chi)\ll q|s|$ and the partial-fraction expansion are available without the functional equation; the contour arguments of §§19–20 only ever use $\sigma\ge 1-c'/\log(qT)$ with $c'$ small.

Formalization Note. InRegion (c/4) q s unfolds to $1-\frac{c/4}{\log(q(|\operatorname{Im}s|+2))}\le\operatorname{Re}s$; analyticOrderNatAt (LFunction χ) z is the order $m_z$ of the zero; the sum over $E$ is a finite sum over the (at most one-element) set $E$ (∑ᶠ); the if χ = 1 then 1/(s-1) else 0 term is $\delta_\chi/(s-1)$. Both conclusions are stated with a single constant $C$ for convenience.