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On sums of Rudin-Shapiro coefficients II

Brillhart, John and Erdős, Paul and Morton, Patrick (1983) On sums of Rudin-Shapiro coefficients II. Pacific Journal of Mathematics, 107 (1). pp. 39-69. ISSN 0030-8730.

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Let {a(n)} be the Rudin-Shapiro sequence, and let s(n) = ∑a(k) and t(n) = ∑(-1)k a(k). In this paper we show that the sequences {s(n)/√n} and {t(n)/√n} do not have cumulative distribution functions, but do have logarithmic distribution functions (given by a specific Lebesgue integral) at each point of the respective intervals [√3/5, √6] and [0, √3]. The functions a(x) and s(x) are also defined for real x ≥ 0, and the function [s(x) – a(x)]/√x is shown to have a Fourier expansion whose coefficients are related to the poles of the Dirichlet series ∑a(n)/n, where Re τ > ½.

Item Type:Article
Additional Information:© 1983 Pacific Journal of Mathematics. Received January 13, 1981. We would like to thank Igor Mikolic-Torreira for carrying out the computations in Table 1 (§6), and Richard Blecksmith for providing us with the graphs in Figure 1 (§4). We are also grateful to A.J.E.M. Janssen for his remarks concering several of our proofs.
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Usage Policy:No commercial reproduction, distribution, display or performance rights in this work are provided.
ID Code:2321
Deposited By: Tony Diaz
Deposited On:27 Mar 2006
Last Modified:26 Dec 2012 08:48

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