Fixed \(\alpha>0\), consider the sum \[S(N;\alpha)
=
\sum_{n=1}^N
e^{2\pi i\alpha\sqrt{n}}.\] When plotting 1000 partial sums for some selected values of \(\alpha\), we get the figures:
This video (avi) gives
an animation when \(\alpha\) varies in [0.05,3].
Here there is a PDF
document with other images and code.
This is related to the research paper:
- [LINK] F.
Chamizo and D. Raboso.
Van der Corput method and optical illusions.
Indag. Math. (N.S.), 26(5):723-735, 2015.
For more elementary material see
this
or the
talk
in the
Rutgers experimental mathematics seminar.
