On sums of Rudin-Shapiro coefficients. II

John David Brillhart, Paul Erdős and Richard Patrick Morton

Vol. 107 (1983), No. 1, 39–69

DOI: 10.2140/pjm.1983.107.39

Abstract

Let {a(n)} be the Rudin-Shapiro sequence, and let s(n) = Σ_k=0ⁿa(k) and t(n) = Σ_k=0ⁿ(−1)^ka(k). In this paper we show that the sequences {s(n)∕} and {t(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 [,] and [0,]. The functions a(x) and s(x) are also defined for real x ≥ 0, and the function [s(x) − a(x)]∕ is shown to have a Fourier expansion whose coefficients are related to the poles of the Dirichlet series Σ_n=1^∞a(n)∕n^τ, where Re τ > .

Mathematical Subject Classification 2000

Primary: 11N60

Secondary: 11N99

Milestones

Received: 13 January 1981

Published: 1 July 1983