Pulsating waves of zeta function Below is an animation of the partial sums of $\operatorname{li}x-2\Re\sum_{k=1}^{N}\operatorname{Ei}(\rho_k \log x)-\log2+\int_{x}^{\infty}\dfrac{\text{d}t}{t(t^2-1)\log t},$ for $1\leq N\leq100$ between $2\leq x\leq3$: !enter image description here Clearly, $k$ can only increase in integer steps, but it seems as though there should be a smooth function (though one is not known as far as I know). The amplitude appears to pulsate with rigid regularity in between peak/trough increases. This may be more of a physics question, but I was wondering whether examples of this phenomenon were common and known elsewhere, and where I might find examples of smooth functions of this kind.

This is essentially Gibbs Phenomenon. See this paper: "Experiments With Zeta Zeros and Perron's Formula", specifically section 4 and 5, pages 4-8.

A very heuristic answer is that you're converging to what's essentially a step function: the (Riemann) prime counting function. If you look at the logorithmic integral $\mbox{Li}(x^{\rho_k})=\int_0^{x^{\rho_k}}\frac{1}{\log(x)}dx$, and conjecture that $\rho_k=1/2+\sigma_k i$, for $\sigma_k$ imaginary, then your $x^{\rho_k}$ oscillates wildly, making the logorithmic integral also oscillate wildly, which is similar to Gibbs phenomenon for fourier expansion of a step function.

I'm not an expert in this area but usually people "deal" with Gibb's phenomenon by using Fourier series that converge faster, i.e. something like Fejer summation. In this case people have (hopefully) found considerably more accurate summations, whose oscillations die off much faster.