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)∕} 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 Σn=1∞a(n)∕nτ, where
Reτ > .