The Möbius Function and the Möbius Transform

  • Manfred R. Schroeder
Part of the Springer Series in Information Sciences book series (SSINF, volume 7)

Abstract

After Euler’s totient function, the Möbius function (named after the Möbius of strip fame) is one of the most important tools of number theory. It allows us to invert certain number-theoretic relations. In a sense, if we liken summation over divisors to integration, then taking the Möbius function is like differentiating. It is defined as follows:
$$ \mu (n): = \left\{ {\begin{array}{*{20}{c}} {1 {\kern 1pt} for {\kern 1pt} n = 1} \\ {0{\kern 1pt} if{\kern 1pt} n {\kern 1pt} is {\kern 1pt} divisible {\kern 1pt} by {\kern 1pt} a {\kern 1pt} square} \\ {{{( - 1)}^K}{\kern 1pt} if{\kern 1pt} n{\kern 1pt} {\kern 1pt} is{\kern 1pt} the{\kern 1pt} product{\kern 1pt} of{\kern 1pt} k{\kern 1pt} distinct{\kern 1pt} {\kern 1pt} primes} \end{array}} \right. $$
(20.1)
Thus, µ(n)≠ 0 only for 1 and squarefree integers. The Möbius function is multiplicative:\( \mu (mn) = \mu (m)\mu (n){\kern 1pt} if(m,n) = 1 \) Also,
$$ \mu (mn) = 0{\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} if(m,n) >1 $$
(20.2)
because then mn is not squarefree.

Keywords

Convolution 

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References

  1. 20.1
    L. H. Hua: Introduction to Number Theory (Springer, Berlin, Heidelberg, New York 1982 )Google Scholar
  2. 20.2
    T. M. Apostol: Introduction to Analytic Number Theory (Springer, Berlin, Heidelberg, New York 1976 )Google Scholar
  3. 20.3
    G. H. Hardy, E. M. Wright: An Introduction to the Theory of Numbers, 4th ed.( Clarendon, Oxford 1960 )MATHGoogle Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 1984

Authors and Affiliations

  • Manfred R. Schroeder
    • 1
    • 2
  1. 1.Drittes Physikalisches InstitutUniversität GöttingenGöttingenFed. Rep. of Germany
  2. 2.Acoustics Speech and Mechanics ResearchBell LaboratoriesMurray HillUSA

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