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**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) = **sum**_{k = 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) = **lim**_{k ––> oo}(s(4^{k} x)/\sqrt{a^{k} 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}**sum**_{n = -oo}^{oo}c_{n}x^{\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) = **sum**_{n = 1}^{oo}a(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.

11A63 Radix representation

**Keywords: ** Rudin-Shapiro sums; binary representations; Fourier series; Dirichlet series; logarithmic distribution

**Citations: ** Zbl.371.10009

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