On sums of Rudin-Shapiro coefficients. II
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 0371.10009).] The properties of these sums are developed further by introducing the limit function \[ \lambda(x)=\lim_{k\to\infty}(s(4^k x)/\sqrt{a^k x}), x>0, \] which turn out to be a continuous function from \((0,\infty)\) onto the interval \(\left[\sqrt{(3/5)},\sqrt6\right]\) and which satisfies the equation \(\lambda(4x)=\lambda(x)\). this function is used to represent \(s(x)\) as a logarithmic Fourier series: \[ s(x)=\sqrt{x}\sum_{n=-\infty}^{\infty}c_nx^{\pi n/\log2}+a(x), x>0, \] 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 \(\eta(\tau)=\sum_{n=1}^{\infty}a(n)n^{-\tau}\) 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 \(\left[\sqrt{(3/5)},\sqrt6\right]\), but that the cumulative distribution function to this sequence does not exist.
scientific article; zbMATH DE number 3735920