On sums of Rudin-Shapiro coefficients II

John Brillhart, P. Erdős, Patrick Morton

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Abstract

Let {a(n)} be the Rudin-Shapiro sequence, and let s(n) = Σk=0 na(k) and t(n) = Σk=0 n(-1)ka(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) sore 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 Σn=1 a(n)/nτ where Re τ > 1/2.

Original languageEnglish
Pages (from-to)39-69
Number of pages31
JournalPacific Journal of Mathematics
Volume107
Issue number1
Publication statusPublished - 1983

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ASJC Scopus subject areas

  • Mathematics(all)

Cite this

Brillhart, J., Erdős, P., & Morton, P. (1983). On sums of Rudin-Shapiro coefficients II. Pacific Journal of Mathematics, 107(1), 39-69.