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Difference between revisions of "Möbius series"

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====Comments====
 
====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 the convolution product, cf. (the editorial comments to) [[Möbius function]] and [[Multiplicative arithmetic function]].
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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
 
The term "Möbius series" is also applied to the summatory function of the Möbius function

Revision as of 17:11, 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
[2] I.M. Vinogradov, "Elements of number theory" , Dover, reprint (1954) (Translated from Russian)
[3] K. Prachar, "Primzahlverteilung" , Springer (1957)


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=38742
This article was adapted from an original article by B.M. Bredikhin (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article