Ikehara's Tauberian theorem for Dirichlet series states that if $$F(s)=\sum_{1}^{\infty}\frac{f(n)}{n^s}$$ with $f(n)\geq 0$ is such that $$F(s)=\frac{G(s)}{s-1}+H(s)$$ for $\sigma>1$ with $G,H$ analytic and $G(1)\neq 0$, then the continuity of $F(1+it)$, $t\neq 0$, implies that $$\frac{1}{x}\sum_{n\leq x}f(n)\sim G(1).$$

I am wondering if there is a converse to this statement, i.e. Do we know that if $F(1+it)$ has other discontinuities at say $t=\pm a_i$, $i=1,2,...$, then $$\frac{1}{x}\sum_{n\leq x}f(n)\not\sim G(1)?$$

If not, I can imagine that there is a converse under stricter assumptions on $F$, say if we assume $F(s)$ is analytic in a neighbourhood of every point of the line $\sigma=1$ except at $t=0$ and $t=\pm a_i$, or under stricter assumptions on the $a_i's$, then $$\frac{1}{x}\sum_{n\leq x}f(n)\not\sim G(1).$$ Do such theorems exist?