I was watching 3Blue1Brown visualizations of the converging values of Riemann Zeta function $\zeta(s)$. From 6:41 forward he shows that each how each series $\sum{\frac{1}{n^{s}}}$ maps into a spiral (vector sum) in the complex plane for $\sigma>1$. The focus is the image $\zeta(s)$. The real part ($Re(s), \sigma)$ determines each vectors length while ($Im(s),t$) determines angles.
After sketching some conditions, I am convinced that the vector sum follow patterns similar to Andronov-Hopf bifurcations after $\sigma$ in a discrete dynamical system.
Negative even values ($\sigma=-2n$) determine a limit cycle related to $exp$ period (rotations over the real line on the Riemann Sphere). Positive real part ($\sigma>1$) determines stable nodes (the spirals from 3B1B video). The higher $\sigma$ is, more convoluted is the spiral, hence $\lim_{\sigma\rightarrow \infty}{\zeta(\sigma+it)=1}$. The critical strip $0 < \sigma < 1$ also contains stable focuses, fractional exponents determine sums with fixed points at the real line.For $\sigma = \frac{1}{2}$, the stable focus is exactly at the real line. One can plot values of $\zeta$ near $\sigma=\frac{1}{2}$ to check the translation of values in the irregular curve representing $Re(\zeta(s))$.
I came here to ask for directions. Am I just another applied mathematician losing time? Is this a valid line of research?
I did not start a formalization, but I was thinking of the following possibilities: (1) Working with orbits on vector fields and then extending the results to the analytic continuation of $\zeta$; (2) Working directly on the dynamical properties of the analytic continuation (e.g. eigenvalues, flow $\phi$ and Liapounoff stability); (3) Working with differential geometry.
For $\Re(s) > 1$ and by analytic continuation for $\Re(s) > 0$ $$\sum_{n=1}^{N-1} n^{-s} = \zeta(s) -\sum_{n=N}^\infty n^{-s} \\ =\zeta(s) - \int_N^\infty x^{-s}dx-\sum_{n=N}^\infty (n^{-s}-\int_n^{n+1}x^{-s}dx)\\= \zeta(s)- \frac{N^{1-s}}{s-1} -\sum_{n=N}^\infty \int_n^{n+1} \int_n^x st^{-s-1}dtdx\\= \zeta(s)- \frac{N^{1-s}}{s-1} + O( \sum_{n=N}^\infty\int_n^{n+1} |s x^{-s-1}|dx)\\ = \zeta(s)- \frac{N^{1-s}}{s-1} + O(s \int_N^\infty x^{-\Re(s)-1}dx)\\ = \zeta(s)- \frac{N^{1-s}}{s-1} + O(\frac{s}{\Re(s)}N^{-s})$$ and if you plot $ \frac{N^{1-s}}{s-1}$ for $\Re(s) >1, s \not \in \Bbb{R}$ you'll get a spiral converging to $0$.
If you substract the main term $\frac{N^{1-s}}{s-1} $ then the asymptotic of the second term isn't more mysterious : it is found following quite the same method, which leads to the Euler McLaurin summation formula. What is mysterious is the value of $\sum_{n =1}^N n^{-s} $ for $N$ of "middle size" $N \approx |\Im(s)|^r$.