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
| dc.creator | Auil, F. | |
| dc.date | 2004-08-06 | |
| dc.date.accessioned | 2026-07-07T05:11:05Z | |
| dc.date.available | 2026-07-07T05:11:05Z | |
| dc.description | 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. | |
| dc.description | 9 pages, no figures | |
| dc.identifier | https://arxiv.org/abs/math/0408093 | |
| dc.identifier | http://arxiv.org/abs/math/0408093 | |
| dc.identifier.uri | http://salesiana.dossiersoluciones.com/handle/123456789/72128 | |
| dc.subject | Number Theory | |
| dc.title | 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 | |
| dc.type | text |