# A Generalized Wiener-Ikehara Theorem with multiple poles on the line

One version of the Wiener-Ikehara Theorem says that if $$f(s) = \sum \frac{a(n)}{n^s}$$ is a Dirichlet series with nonnegative coefficients that converges absolutely for $\text{Re}(s) > 1$ and with meromorphic continuation to $\text{Re}(s) \geq 1$ with only pole given by a simple pole at $s = 1$ with residue $R \geq 0$, then $$\sum_{n \leq X} a(n) = RX + o(X).$$

I'm thinking about whether there is a variant which considers more than one pole. For instance, suppose $g(s)$ is a Dirichlet series with coefficients $b(n)$ and as above, except with simple poles at $1, 1+i, 1-i$ with residues $R_1, R_i, R_{-i}$. I feel that we should be able to conclude something like $$\sum_{n \leq X} b(n) = R_1 X + \frac{R_i}{1+i}X^{1+i} + \frac{R_{-i}}{1-i}X^{1-i} + o(X). \tag{1}$$

If we knew more analytic information about $f(s)$ then we can prove this in other ways. For instance, if we hypothetically knew that $f(s)$ had exponential decay in vertical strips for $0 < \text{Re}(s) < 1$, then we could apply a Perron-style formula and make equation $(1)$ certain. But part of the beauty of Wiener-Ikehara is that we don't need that extra information (and sometimes, as in the case of the zeta function, we barely have much more information).