A Sequence of Beurling Functions Related to the Natural Approximation B_{n} Defined by an Iterative Construction Generating Square-Free Numbers k and the Value of the Mobius Function at k

Abstract

We construct iteratively a sequence of numbers k_{n} and Beurling functions A_{n} converging pointwise to -1 in [0,1]. We prove results which seems to suggest that each A_{n} is equal to a well known approximating sequence of functions denoted in literature by B_{n}; see ref. [2]. We conjecture that a sufficient condition for this equality is that the set of the k_{n}'s be equal to the set of square-free numbers. Numerical evidence seems to support both conjectures. Anyway, we think that these sequences are interesting by itself because our construction not only generates square-free (hence prime) numbers k, but also the value of the Mobius function at k. Our definition is independent of the square-free numbers and the Mobius function, with the k_{i}'s arising as discontinuity points of the A_{i}'s. As for the case of B_{n}, we prove that sequence A_{n} is not convergent to -1 in L^{2}([0,1],dx). Consequently, we focus our analysis not on L^{2} norm analysis but other integral properties. This procedure seems to be useful to elucidate the lack of L^{2} convergence for step Beurling functions.