View source for Möbius series

A series of functions of the form

<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m064/m064300/m0643001.png" /></td> <td valign="top" style="width:5%;text-align:right;">(*)</td></tr></table>

These series were investigated by A. Möbius [[#References|[1]]], who found for a series (*) the inversion formula

<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m064/m064300/m0643002.png" /></td> </tr></table>

where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m064/m064300/m0643003.png" /> is the [[Möbius function|Möbius function]]. Möbius considered also inversion formulas for finite sums running over the divisors of a natural number <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m064/m064300/m0643004.png" />:

<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m064/m064300/m0643005.png" /></td> </tr></table>

Another inversion formula: If <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m064/m064300/m0643006.png" /> is a totally-multiplicative function (cf. [[Multiplicative arithmetic function|Multiplicative arithmetic function]]) for which <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m064/m064300/m0643007.png" />, and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m064/m064300/m0643008.png" /> is a function defined for all real <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m064/m064300/m0643009.png" />, then

<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m064/m064300/m06430010.png" /></td> </tr></table>

implies

<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m064/m064300/m06430011.png" /></td> </tr></table>

====References====
<table><TR><TD valign="top">[1]</TD> <TD valign="top">  A. Möbius,   "Ueber eine besondere Art der Umkehrung der Reihen"  ''J. Reine Angew. Math.'' , '''9'''  (1832)  pp. 105–123</TD></TR><TR><TD valign="top">[2]</TD> <TD valign="top">  I.M. Vinogradov,   "Elements of number theory" , Dover, reprint  (1954)  (Translated from Russian)</TD></TR><TR><TD valign="top">[3]</TD> <TD valign="top">  K. Prachar,   "Primzahlverteilung" , Springer  (1957)</TD></TR></table>



====Comments====
All these (and many other) inversion formulas follow from the basic property of the Möbius function that it is the inverse of the unit arithmetic function <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m064/m064300/m06430012.png" /> under the convolution product, cf. (the editorial comments to) [[Möbius function|Möbius function]] and [[Multiplicative arithmetic function|Multiplicative arithmetic function]].