Explore

Things to turn over in your hands. Each page takes one function — or one family of functions — and wires its every constant to a dial, so the question is not "what does this do" but "what happens when I move this".

The guides explain; these pages let you poke. Where a claim here is better made in prose, it links back.

• The two-argument zeta — the Hurwitz zeta $\zeta(s, a)$ and its generalized cousin, by Euler–Maclaurin: a surface, the closed forms, and where the two conventions part ways.
  • ζ on the GPU: a phase portrait — the whole kernel compiled to a WebGPU shader, colouring the complex $a$-plane in real time.
• The Lerch transcendent — $\Phi(z, s, a)$, the one function above the zeta/polylog/eta family, with a GPU domain-colouring of its disk of convergence.
• The polylog and the polygamma — $\operatorname{Li}_s(z)$ and $\psi^{(m)}(z)$, both of them the Lerch series or the Hurwitz kernel in disguise, and two GPU portraits that look nothing alike.
• Fractals — the Mandelbrot and Julia sets of $z \mapsto z^2 + c$, coloured by the $n$-th iterate itself rather than an escape count, with $n$ and $c$ on dials.
• Quadratic primes — the primes of every ring of quadratic integers on a lattice, field after field.
• Complex bases — numbers written in a Gaussian or Eisenstein base, every numeral a tile: the twindragon, the Gosper island, and systems of your own.

How these are built

Every dial on these pages is a <notatio-manipulate> wrapped around a plotting component, with the swept constants written as named wildcards (_s, _a) in the expression itself. Nothing here is bespoke: the same components are documented in the component reference and exercised one at a time in the playground. If a page here suggests a rig you want, you can build it.