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On finite pseudorandom binary sequences. III: The Liouville function. I. (English) Zbl 0920.11053

This is the third part of an extensive investigation on pseudorandom properties of arithmetic sequences initiated by C. Mauduit and A. Sarközy, see [Acta Arith. 82, 365–377 (1997; Zbl 0886.11048) and J. Number Theory 73, 256–276 (1998; Zbl 0916.11047)]. The authors mainly consider the “correlation measure of order \(k\)” of binary sequences over \(\{-1,+1\}\).
In part I of their work they were concerned with the sequence \(e_n=({n\over p})\), \(n=1,\dots,N\) of Legendre symbols, \(p\) a prime number. In the present paper the authors investigate the sequence \(\lambda(n)=(-1)^{\Omega (n)}\) and \(\gamma(n)=(-1)^{(\omega(n)}\), where \(\omega(n)\) denotes the number of distinct prime factors of \(n\) and \(\Omega(n)\) the number of prime factors of \(n\) counted with multiplicity. The authors obtain estimates for a “well-distribution measure” and for the “correlation measure of order \(k\)”. Moreover, they study the complexity of the sequences and a connection between correlation and complexity.
Reviewer: R.F.Tichy (Graz)

MSC:

11K45 Pseudo-random numbers; Monte Carlo methods
65C10 Random number generation in numerical analysis
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