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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
##### Keywords:
Riemann’s nondifferentiable function; selfsimilarity