Let $D(s) = \sum_{n=1}^\infty a_n n^{-s}$ be a Dirichlet series with $a_n ≥ 0$ and abscissa of convergence $\sigma_a = 1$. Further, we assume that $D(s)$ is holomorphic in each point $\Re(s) = 1$ except a pole of order $k > 0$ in $s = 1$ and that $D(s)$ possesses a meromorphic continuation on some half-plane $\Re(s) > 1 - \epsilon_0$ with some $\epsilon_0 > 0$.

My question is, whether under the above assumptions we can already say that

$$ \sum_{n ≤ x} a_n = xP(\log x) + o(x)$$

with some polynomial $P$ with degree $k-1$. Unfortunately I am not able to exclude other poles in the strip $1 > \Re(s) > 1 - \epsilon_0$.

In the simple case $k=1$ the question follows by the Wiener-Ikehara theorem, since then we have $\sum_{n ≤ x} a_n \sim Cx$ with $C = \mathrm{res}_{s=1}D(s)$.

Thank you!