I already asked this question here

but I did not get a satisfying answer, so I put it here. I would like to estimate (upper and lower bounds for real and imaginary parts) the following sums and series:

$$ f_x(t)=\sum_{n=1}^x\exp(\pi i t/n) $$

$$ g_x(t)=\sum_{n=-x}^x\exp(\pi i t/n),\qquad n\neq0 $$ by some (complex) functions of $t$ or $t,x$, and the same question for their absolute values: $|f_x(t)|,|g_x(t)|$?

Thanks!

P.S. I wrote some code in "**Mathematica**" as follows:

```
f[t_] := Sum[Exp[Pi I t/n], {n, 1, x}];
x = 200;
Plot[{Abs[f[t]], x}, {t, 1, x}]
```

and I get something like

Of course $x$ is upper bound, but I am looking for asymptotic functions like the "red" ones in the picture that are close to $f_x(t)$.

P.S. I corrected some divergence parts. I added the "red" curve in "paint".