An algorithm to generate square-free numbers and to compute the Möbius function

We introduce an algorithm that iteratively produces a sequence of natural numbers ki and functions bi defined in the interval [0,+∞). The number ki+1 arises as the first point of discontinuity of bi above ki. We prove that (1) the algorithm deterministically produces the complete sequence of square-free numbers ki in increasing order, and (2) the value of the Möbius function μ(ki) can be evaluated as bi(ki+1)−bi(ki). Our analysis is closely related to the Nyman–Beurling approach to the Riemann hypothesis.