Vol. 107, No. 1, 1983

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On sums of Rudin-Shapiro coefficients. II

John David Brillhart, Paul Erdős and Richard Patrick Morton

Vol. 107 (1983), No. 1, 39–69
Abstract

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

Mathematical Subject Classification 2000
Primary: 11N60
Secondary: 11N99
Milestones
Received: 13 January 1981
Published: 1 July 1983
Authors
John David Brillhart
Paul Erdős
Richard Patrick Morton