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Selfsimilarity of “Riemann’s nondifferentiable function”. (English) Zbl 0760.26009
The author gives a self-contained presentation of the differentiability properties of the function \(f(x):=\sum^ \infty_{n=1}\sin(n^ 2\pi x)/(n^ 2\pi)\) using selfsimilarity of the corresponding theta function. His main result is an asymptotic description of \(\varphi(x):=\sum^ \infty_{n=1}\exp(n^ 2 i\pi x)/(n^ 2 i\pi)\) near rational points leading to an explanation of infinitely many selfsimilarities of the graph of \(f(x)\). However, the fractional dimension is not discussed and picture resolution is very low.
Surprisingly the author takes no notice of important works in this area as H. Queffelec [“Dérivabilité de certaines sommes de séries de Fourier lacunaires”, Thèse, Paris, Orsay (1971)] and M. V. Berry and J. Goldberg [Nonlinearity 1, No. 1, 1-26 (1988; Zbl 0662.10029)].
Reviewer: W.Luther (Aachen)

MSC:
26A27 Nondifferentiability (nondifferentiable functions, points of nondifferentiability), discontinuous derivatives
28A80 Fractals
11A55 Continued fractions
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