# a pathological function of Riemann

The periodic mantissa function $t\mapsto t-\lfloor t\rfloor $ has at each integer value of $t$ a jump (saltus) equal to $-1$, being in these points continuous^{} from the right but not from the left. For every real value $t$, one has

$$ | (1) |

Let us consider the series

$\sum _{n=1}^{\mathrm{\infty}}}{\displaystyle \frac{nx-\lfloor nx\rfloor}{{n}^{2}}$ | (2) |

due to Riemann. Since by (1), all values of $x\in \mathbb{R}$ and $n\in {\mathbb{Z}}_{+}$ satisfy

$$ | (3) |

the series is, by Weierstrass’ M-test, uniformly convergent on the whole $\mathbb{R}$ (see also the p-test). We denote by $S(x)$ the sum function^{} of (2).

The ${n}^{\mathrm{th}}$ term of the series (2) defines a periodic function^{}

$x\mapsto {\displaystyle \frac{nx-\lfloor nx\rfloor}{{n}^{2}}}$ | (4) |

with the period (http://planetmath.org/PeriodicFunctions) $\frac{1}{n}$ and having especially for $$ the value $\frac{x}{n}$. The only points of discontinuity of this function are

$x={\displaystyle \frac{m}{n}}\mathit{\hspace{1em}\hspace{1em}}(m=0,\pm 1,\pm 2,\mathrm{\dots}),$ | (5) |

where it vanishes and where it is continuous from the right but not from the left; in the point (5) this function apparently has the jump $-{\displaystyle \frac{1}{{n}^{2}}}$.

The theorem of the entry one-sided continuity by series implies that the sum function $S(x)$ is continuous in every irrational point $x$, because the series (2) is uniformly convergent for every $x$ and its terms are continuous for irrational points $x$.

Since the terms (4) of (2) are continuous from the right in the rational points (5), the same theorem implies that
$S(x)$ is in these points continuous from the right. It can be shown that $S(x)$ is in these points discontinuous^{} from the left having the jump equal to $-{\displaystyle \frac{{\pi}^{2}}{6{n}^{2}}}$.

## References

- 1 E. Lindelöf: Differentiali- ja integralilasku ja sen sovellutukset III.2. Mercatorin Kirjapaino Osakeyhtiö, Helsinki (1940).

Title | a pathological function of Riemann |
---|---|

Canonical name | APathologicalFunctionOfRiemann |

Date of creation | 2013-03-22 18:34:17 |

Last modified on | 2013-03-22 18:34:17 |

Owner | pahio (2872) |

Last modified by | pahio (2872) |

Numerical id | 9 |

Author | pahio (2872) |

Entry type | Example |

Classification | msc 40A05 |

Classification | msc 26A15 |

Classification | msc 26A03 |

Synonym | example of semicontinuous function |

Related topic | DirichletsFunction |

Related topic | ValueOfTheRiemannZetaFunctionAtS2 |