In this post, the questioner asked about the behavior of a Dirichlet series over Bernoulli polynomials:
$$ \mathcal{B}(k,s) = \sum_{m\geq 1} \frac{B_k(m)}{m^s}. $$
They show that this sum is equal to
$$ \mathcal{B}(k,s) =\sum_{i=0}^k\binom{k}{i}B_i\zeta(i + s - k). $$
which is relatively clean. However, I am curious about the concrete behavior of this latter sum. Is there a good way to approximate $\mathcal{B}(k,s)$, in the limit of large $k$? In particular, what is $\mathcal{B}(k,s)$ asymptotic to in the limit $k \rightarrow \infty$?