For which function $f$ is $1 \ll \sum_{i=1}^{n} i \cdot i^{-f(n)} \ll n$?

Question

I am interested in the expected value of a power-law Distribution. I would like to let the Parameter $f(n)$ depend on $n$ for $n \rightarrow \infty$.

And now I would like to determine $f(n)$ such that

$$ 1\ll \sum_{i=1}^n i\cdot i^{-f(n)} \ll n$$.

For which easy functions $f$ does this hold?

Examples might be $\log(n)$, $n$, $\frac{1}{n}$, $\frac{1}{\log n}$. I have no idea how to solve this sum and I am not sure whether there is a Closed-form term for this. But I am mainly interested in the function values $f$ for which above holds.

And for which functions $f$ does it hold that

$$\sum_{i=1}^n i\cdot i^{-f(n)} \gg 1.$$

I do not necessarily Need a detailed desription of all the functions that satisfy These inequalities. I only Need to have some example functions, at least.

Answer 1

Necessary conditions: $\liminf_{n\to\infty}f(n)\geqslant1$ and $\limsup\limits_{n\to\infty}f(n)\leqslant2$.

Sufficient conditions: $\liminf\limits_{n\to\infty}f(n)\gt1$ and $\limsup\limits_{n\to\infty}f(n)\lt2$.

Examples: every constant $f(n)=\alpha$, provided $1\lt\alpha\lt2$.