Define
\begin{equation}
F(\sigma) = \Re \sum_{h=1}^{\infty} \sum_{n=1}^{\infty} \frac{b_{n, h}}{n^{2\sigma}}(1+h/n)^{-\sigma}\Bigg( \frac{e^{i\log(1+h/n)}-1}{i\log(1+h/n)} \Bigg)
\end{equation} where $\sigma \in \mathbb{R}$ and $|b_{n, h}|\ll \log n$. *What is the minimal real number $c$ such that $F(\sigma)$ has a holomorphic continuation in $\sigma>c$ ?*

Heuristically, it seems to me that $c \leq 1/2$, considering the fact that the inner sum is dominated by terms with small $h/n$, so that convergence of the whole series is assured for $2\sigma>1$.