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m (moved Moebius series to Möbius series over redirect: accented title)
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A series of functions of the form
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A class of functions of the form
 
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$$
<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>
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F_n(x) = \sum_{s=1}^\infty f(x^s) s^{-n} \ .
 
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$$
 
These series were investigated by A. Möbius [[#References|[1]]], who found for a series (*) the inversion formula
 
These series were investigated by A. Möbius [[#References|[1]]], who found for a series (*) the inversion formula
 +
$$
 +
f(x) = \sum_{s=1}^\infty \mu(s) F_n(x^s) s^{-n} \ ,
 +
$$
 +
where $\mu(s)$ is the [[Möbius function]]. Möbius considered also inversion formulas for finite sums running over the divisors of a natural number $n$:
 +
$$
 +
F(n) = \sum_{d | n} f(d) \ ,\ \ \ f(n) = \sum_{d | n} \mu(d) F(n/d) \ .
 +
$$
  
<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>
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Another inversion formula: If $P(n)$ is a [[totally multiplicative function]] for which $P(1) = 1$, and $f(x)$ is a function defined for all real $x > 0$, then
 +
$$
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g(x) = \sum_{n \le x} P(n) f(x/n)
 +
$$
 +
implies
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$$
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f(x) = \sum_{n \le x} \mu(n) P(n) g(x/n) \ .
 +
$$
  
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" />:
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====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 {{DOI|10.1515/crll.1832.9.105}} {{ZBL|009.0333cj}}</TD></TR>
 +
<TR><TD valign="top">[2]</TD> <TD valign="top">  I.M. Vinogradov,  "Elements of number theory" , Dover, reprint  (1954)  (Translated from Russian) {{ZBL|0057.28201}}</TD></TR>
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<TR><TD valign="top">[3]</TD> <TD valign="top">  K. Prachar,  "Primzahlverteilung" , Springer  (1957) {{ZBL|0080.25901}}</TD></TR>
 +
</table>
  
<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>
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====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 $E(n) \equiv 1$ under [[Dirichlet convolution]], cf. (the editorial comments to) [[Möbius function]] and [[Multiplicative arithmetic function]].
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>
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The term "Möbius series" is also applied to the summatory function of the Möbius function
 +
$$
 +
M(x) = \sum_{n \le x} \mu(n) \ .
 +
$$
 +
Mertens conjectured in 1897 that the bound $|M(x)| < \sqrt x$ holds: this would imply the [[Riemann hypothesis]]. Odlyzko and te Riele disproved the Mertens conjecture in 1985.
  
 
====References====
 
====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>
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* Odlyzko, A.M.; te Riele, Herman J.J. "Disproof of the Mertens conjecture"  ''J. Reine Angew. Math.'' '''357''' (1985) 138-160 {{DOI|10.1515/crll.1985.357.138}} {{ZBL|544.10047}}
 
 
  
 
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{{TEX|done}}
====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]].
 

Latest revision as of 17:15, 30 April 2016

A class of functions of the form $$ F_n(x) = \sum_{s=1}^\infty f(x^s) s^{-n} \ . $$ These series were investigated by A. Möbius [1], who found for a series (*) the inversion formula $$ f(x) = \sum_{s=1}^\infty \mu(s) F_n(x^s) s^{-n} \ , $$ where $\mu(s)$ is the Möbius function. Möbius considered also inversion formulas for finite sums running over the divisors of a natural number $n$: $$ F(n) = \sum_{d | n} f(d) \ ,\ \ \ f(n) = \sum_{d | n} \mu(d) F(n/d) \ . $$

Another inversion formula: If $P(n)$ is a totally multiplicative function for which $P(1) = 1$, and $f(x)$ is a function defined for all real $x > 0$, then $$ g(x) = \sum_{n \le x} P(n) f(x/n) $$ implies $$ f(x) = \sum_{n \le x} \mu(n) P(n) g(x/n) \ . $$

References

[1] A. Möbius, "Ueber eine besondere Art der Umkehrung der Reihen" J. Reine Angew. Math. , 9 (1832) pp. 105–123 DOI 10.1515/crll.1832.9.105 Zbl 009.0333cj
[2] I.M. Vinogradov, "Elements of number theory" , Dover, reprint (1954) (Translated from Russian) Zbl 0057.28201
[3] K. Prachar, "Primzahlverteilung" , Springer (1957) Zbl 0080.25901

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 $E(n) \equiv 1$ under Dirichlet convolution, cf. (the editorial comments to) Möbius function and Multiplicative arithmetic function.

The term "Möbius series" is also applied to the summatory function of the Möbius function $$ M(x) = \sum_{n \le x} \mu(n) \ . $$ Mertens conjectured in 1897 that the bound $|M(x)| < \sqrt x$ holds: this would imply the Riemann hypothesis. Odlyzko and te Riele disproved the Mertens conjecture in 1985.

References

How to Cite This Entry:
Möbius series. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=M%C3%B6bius_series&oldid=23420
This article was adapted from an original article by B.M. Bredikhin (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article