Published July 1, 1983
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Journal Article
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On sums of Rudin-Shapiro coefficients II
- Creators
- Brillhart, John
- Erdős, Paul
- Morton, Patrick
Abstract
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.
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.Files
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- Eprint ID
- 2321
- Resolver ID
- CaltechAUTHORS:BRIpjm83.805
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