# 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

 $\displaystyle 0\leqq t\!-\!\lfloor{t}\rfloor<1.$ (1)

Let us consider the series

 $\displaystyle\sum_{n=1}^{\infty}\frac{nx\!-\!\lfloor{nx}\rfloor}{n^{2}}$ (2)

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

 $\displaystyle 0\;\leqq\;\frac{nx\!-\!\lfloor{nx}\rfloor}{n^{2}}\;<\;\frac{1}{n% ^{2}},$ (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

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

with the period (http://planetmath.org/PeriodicFunctions) $\frac{1}{n}$ and having especially for  $0\leqq x<\frac{1}{n}$  the value $\frac{x}{n}$.  The only points of discontinuity of this function are

 $\displaystyle x\;=\;\frac{m}{n}\qquad(m=0,\,\pm 1,\,\pm 2,\,\ldots),$ (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  $\displaystyle-\frac{\pi^{2}}{6n^{2}}$.

## References

• 1 E. Lindelöf: Differentiali- ja integralilasku ja sen sovellutukset III.2.  Mercatorin Kirjapaino Osakeyhtiö, Helsinki (1940).
Title a pathological function of Riemann APathologicalFunctionOfRiemann 2013-03-22 18:34:17 2013-03-22 18:34:17 pahio (2872) pahio (2872) 9 pahio (2872) Example msc 40A05 msc 26A15 msc 26A03 example of semicontinuous function DirichletsFunction ValueOfTheRiemannZetaFunctionAtS2