I'm working with a partial summation in the following form:
$$f(t)=\sum_{n=1}^t r_ne^{i\theta_n}$$
where$$r_n=n^{-x}, \theta_n=y\ln(n)+\pi n$$ This sum is always convergent as $t\to\infty$ when $x\in(0,1)$ and $y\in\mathbb{R}$.
I'm curious as to if there is a way to formulate this into a continuous partial sum for $t\in\mathbb{R}_{>0}$ with no kinks. Considering both $r_n$ and $\theta_n$ are continuous and respectively strictly decreasing and increasing, my intuition tells me it should be possible.
Note: I'm not looking to linearly map between integer values of $t$. I.e; the result should be theoretically differentiated everywhere.
Edit: As Eric Wofsey pointed out, "Any sequence of values at the integers can be interpolated by a smooth function (even by an analytic function)." Hence, indeed I'm looking for an interpolation that is "natural" in some sense. Upon experimenting with different modes of interpolation myself, when split into it's real and imaginary components, the function seems to look very sinusoidal (obviously due to Euler's formula) with the integer valued outputs landing directly between the crests and troughs.