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

2004/08/06 by F. Auil, Auil, F.
Mathematics · #FOS: Mathematics #Number Theory (math.NT) #math.NT

paper · pdf · doi:10.48550/arxiv.math/0408093

9 pages, no figures

arxiv created 2004/08/06 · arxiv updated 2009/12/01

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 L²([0,1],dx). Consequently, we focus our analysis not on L² norm analysis but other integral properties. This procedure seems to be useful to elucidate the lack of L² convergence for step Beurling functions.

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