## Zentralblatt MATH

Publications of (and about) Paul Erdös

Zbl.No:  469.10034
Autor:  Brillhart, John; Erdös, Paul; Morton, Patrick
Title:  On sums of Rudin-Shapiro coefficients. II. (In English)
Source:  Pac. J. Math. 107, 39-69 (1983).
Review:  This paper is an extension of previous work by the first and third authors on the Rudin-Shapiro sums s(x) = sumk = 0[x]a(k), where a(k) is defined to be plus of minus on according as the number of pairs of consecutive 1's in the binary represantation of k is even or odd. [See Ill. J. Math. 22, 126-148 (1978; Zbl 371.10009).] The properties of these sums are developed further by introducing the limit function

\lambda(x) = limk ––> oo(s(4k x)/\sqrt{ak x}), x > 0,

which turn out to be a continuous function from (0,oo) onto the interval [\sqrt{(3/5)},\sqrt6] and which satisfies the equation \lambda(4x) = \lambda(x). this function is used to represent s(x) as a logarithmic Fourier series:

s(x) = \sqrt{x}sumn = -oooocnx\pi n/ log2+a(x), x > 0,

Where a(x) is an explicit bounded function of the digits of x to the base 4, which extends a(k) to the set of positive reals. The series (1) is shown to converge for almost all positive real numbers; in particular, it converges for all x > 0 which are normal to the base 4. It turns out that \lambda(x) is non-differentiable on this same set. This is then used to show that the Dirichlet series \eta(\tau) = sumn = 1ooa(n)n-\tau has a meromorphic continuation to the whole complex plane with infinitely many poles. Finally, \lambda(x) is used to prove that the sequence \left{\frac{s(n)}{\sqrt n}\right}n \geq 1 has a logarithmic sistribution function on the interval [\sqrt{(3/5)},\sqrt6], but that the cumulative distribution function to this sequence does not exist.
Classif.:  * 11B83 Special sequences of integers and polynomials
11K65 Arithmetic functions (probabilistic number theory)
11K16 Normal numbers, etc.