# 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

Number Theory Prime Power Prime Divisor Dirichlet Series Inversion Formula
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

## 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 )