On the nonderivability of periodic functions associated with certain summation formulas. (Sur la non-dérivabilité de fonctions périodiques associées à certaines formules sommatoires.) (French) Zbl 0869.11019

Graham, Ronald L. (ed.) et al., The mathematics of Paul Erdős. Vol. I. Berlin: Springer. Algorithms Comb. 13, 117-128 (1997).
In the present paper several digital sums are considered. For instance, H. Delange [Enseign. Math., II. Ser. 21, 31-47 (1975; Zbl 0306.10005)] proved the summation formula \[ \sum_{m<n} \sigma_q(m) ={q-1 \over 2\log q} n\log n+nG_q \left({\log n \over \log q} \right), \] where \(\sigma_q\) denotes the sum-of-digits function in base \(q\). Delange further showed that \(G_q\) is a continuous periodic function of period 1 which is nowhere differentiable. Summation formulae of this kind were extended to various other digital problems such as Rudin-Shapiro sequences and related problems (see for instance J. Brillhart, P. Erdös and P. Morton [Pac. J. Math. 107, 39-69 (1983; Zbl 0505.10029)], K. Stolarsky [SIAM J. Appl. Math. 32, 717-730 (1977; Zbl 0355.10012)], J. M. Dumont and A. Thomas [J. Number Theory 39, 351-366 (1991; Zbl 0736.11007)]. A systematic treatment of digital problems of this kind is due to E. Cateland [Thèse, Univ. Bordeaux (1992)]. An approach to such digital summation formulae via the Mellin-Perron summation formula was established by P. Flajolet, P. Grabner, P. Kirschenhofer, H. Prodinger and R. F. Tichy [Theor. Comput. Sci. 123, 291-314 (1994; Zbl 0788.44004)]. Most of these papers after Delange do not contain proofs for the non-differentiability of the corresponding remainder functions.
The present paper completes the above mentioned investigations and gives proofs for the non-differentiability of the remainder functions. More precisely, Lipschitz conditions for the remainder functions are shown (in the case of Rudin-Shapiro sequences and in the case of the Newman-Coquet problem) so that the Hausdorff dimensions of the graphs can be computed. Furthermore the present paper contains a valuable list of references.
For the entire collection see [Zbl 0857.00027].
Reviewer: R.F.Tichy (Graz)


11B83 Special sequences and polynomials
11K16 Normal numbers, radix expansions, Pisot numbers, Salem numbers, good lattice points, etc.
11A63 Radix representation; digital problems