For a long time, I’ve been fascinated by the apparent randomness of the Möbius function — a sequence that seems to oscillate between +1, 0, and –1 with no visible pattern. But what if there is a hidden structure behind this chaos — one that can be seen, not just calculated?
This idea led me to define a new arithmetic function Δₖ(n). It arises naturally when integers are placed on what I call the number plane — a geometric representation where every number occupies a position determined by its modular residues. The function simply measures the asymmetry in the distribution of coprime residues around a chosen axis. For prime values of k, Δₖ(n) turns out to take only two values for primes — 0 and –2 — dividing them cleanly into two modular classes.
What’s interesting is that the sign of Δₖ(n) correlates with the sign of the Möbius function μ(n) with unexpectedly high accuracy. For square-free numbers, the agreement reaches over 93% up to 10⁶, and remains stable even when several Δₖ functions are combined. In other words — purely geometric properties of the number plane can mirror the algebraic structure of Möbius, without any use of factorisation.
This discovery doesn’t solve the great mysteries of the Möbius or Riemann world — but it opens a small, new window into how order might emerge from arithmetic chaos.
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