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Riemann Function

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Continuous Nowhere Differentiable Functions

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Abstract

The aim of this chapter is to discuss the problem of differentiability of the classical Riemann function

$$\displaystyle{\boldsymbol{R}(x):=\sum _{ n=1}^{\infty }\frac{\sin (\pi n^{2}x)} {n^{2}},\quad x \in \mathbb{R}.}$$

To get some feeling of the behavior of \(\boldsymbol{R}\) see Fig. 13.1.

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References

  1. P. Bois-Reymond, Versuch einer Classification der willkürlichen Functionen reeller Argumente nach ihren Aenderungen in den kleinsten Intervallen. J. Reine Angew. Math. 79, 21–37 (1874)

    MathSciNet  MATH  Google Scholar 

  2. J. Gerver, The differentiability of the Riemann function at certain rational multiples of π. Am. J. Math. 92, 33–55 (1970)

    Google Scholar 

  3. J.Gerver, More on the differentiability of the Riemann function. Am. J. Math. 93, 33–41 (1971)

    Article  MathSciNet  Google Scholar 

  4. G.H. Hardy, Weierstrass’s non-differentiable function. Trans. Am. Math. Soc. 17, 301–325 (1916)

    MathSciNet  MATH  Google Scholar 

  5. S. Itatsu, Differentiability of Riemann’s function. Proc. Jpn. Acad. Ser. A Math. Sci. 57, 492–495 (1981)

    Article  MathSciNet  Google Scholar 

  6. E. Mohr, Wo ist die Riemannsche Funktion \(\sum \limits _{n=1}^{\infty }\frac{\sin n^{2}x} {n^{2}}\) nichtdifferenzierbar? Ann. Mat. Pura Appl. 123, 93–104 (1980)

    Article  MathSciNet  Google Scholar 

  7. A. Smith, The differentiability of Riemann’s functions. Proc. Am. Math. Soc. 34, 463–468 (1972)

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  8. A. Smith, Correction to: “The differentiability of Riemann’s function” [Proc. Am. Math. Soc. 34 (1972)]. Proc. Am. Math. Soc. 89, 567–568 (1983)

    Google Scholar 

  9. K. Weierstrass, Mathematische Werke (Mayer & Müller, Berlin, 1895)

    MATH  Google Scholar 

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Author information

Authors and Affiliations

  1. Institute of Mathematics, Jagiellonian University, Kraków, Poland

    Marek Jarnicki

  2. Insitute for Mathematics, Carl von Ossietzky University Oldenburg, Oldenburg, Germany

    Peter Pflug

Authors
  1. Marek Jarnicki
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  2. Peter Pflug
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Jarnicki, M., Pflug, P. (2015). Riemann Function. In: Continuous Nowhere Differentiable Functions. Springer Monographs in Mathematics. Springer, Cham. https://doi.org/10.1007/978-3-319-12670-8_13

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