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On sums of Rudin-Shapiro coefficients. II

1983; Mathematical Sciences Publishers; Volume: 107; Issue: 1 Linguagem: Inglês

10.2140/pjm.1983.107.39

1945-5844

John Brillhart, P. Erdős, Richard Morton,

Analytic Number Theory Research

Let {a(n)} be the Rudin-Shapiro sequence, and let s(n) = Σ£ = o a ( k ) and t(n) = I" k=0 (-\) k a(k).In this paper we show that the sequences {s{n)/ Jn) and {t{n)/ Jn) 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 [γ/3/5, yfβ] and [0, V^] 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 Σ~=, a(n)/n\ where Re τ > {.