A trivial example might be the harmonic series, where $\sum_{n\leq x} \frac 1 n \sim \ln x$ and asymptotically $n\sim n$.
Another example, the $n$-th prime is $p_n \sim n\ln n$ and $\sum_{p\leq x} \frac 1 p \sim \ln \ln x$.
The questions is, what can we say about the growth rate of $\sum _{a\leq x} \frac 1 a$ if we know the asymptotics for $a_n$?