Estimate of $\sum_{n=1}^x\exp(\pi i t/n)$ and $\sum_{n=-x}^x\exp(\pi i t/n)$?

Question

I would like to estimate (upper and lower bounds for real and imaginary parts) the following sums and series:

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

$$ g_x(t)=\sum_{n=-x}^x\exp(\pi i t/n) $$ 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".

Answer 1

Notice that

$$\lim_{n\to\pm\infty}e^{\pi it/n}=1$$

So that

$$f_x(t)\sim\sum_{n=1}^x1=x\implies f(t)\to\infty$$

$$g_x(t)\sim\sum_{n=-x}^x1=2x+1\implies g(t)\to\infty$$