Möbius function: Difference between revisions
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In [[number theory]], the '''Möbius function''' μ(''n'') is an [[arithmetic function]] which takes the values -1, 0 or +1 depending on the [[prime factorisation]] of its input ''n''. | In [[number theory]], the '''Möbius function''' μ(''n'') is an [[arithmetic function]] which takes the values -1, 0 or +1 depending on the [[prime factorisation]] of its input ''n''. | ||
If the [[positive integer]] ''n'' has a repeated prime factor then μ(''n'') is defined to be zero. If ''n'' is [[square-free]], then μ(''n'') = +1 if ''n'' has an even number of prime factors and -1 if ''n'' has an odd number of prime factors. | If the [[positive integer]] ''n'' has a repeated prime factor then μ(''n'') is defined to be zero. If ''n'' is [[square-free integer|square-free]], then μ(''n'') = +1 if ''n'' has an even number of prime factors and -1 if ''n'' has an odd number of prime factors. | ||
The Möbius function is [[multiplicative function|multiplicative]], and hence the associated [[formal Dirichlet series]] has an [[Euler product]] | The Möbius function is [[multiplicative function|multiplicative]], and hence the associated [[formal Dirichlet series]] has an [[Euler product]] | ||
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giving the '''Möbius inversion formula''' | giving the '''Möbius inversion formula''' | ||
:<math>f(n) = \sum_{d|n} \mu(d) | :<math>f(n) = \sum_{d|n} \mu(d)g(n/d) .\,</math> | ||
A useful special case is the formula | |||
:<math>\sum_{d|n} \mu(d) = 1 \mbox{ if } n=1 \mbox{ and } 0 \mbox{ if } n>1. \,</math> | |||
==Mertens conjecture== | ==Mertens conjecture== |
Latest revision as of 04:58, 10 December 2008
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In number theory, the Möbius function μ(n) is an arithmetic function which takes the values -1, 0 or +1 depending on the prime factorisation of its input n.
If the positive integer n has a repeated prime factor then μ(n) is defined to be zero. If n is square-free, then μ(n) = +1 if n has an even number of prime factors and -1 if n has an odd number of prime factors.
The Möbius function is multiplicative, and hence the associated formal Dirichlet series has an Euler product
Comparison with the zeta function shows that formally at least .
Möbius inversion formula
Let f be an arithmetic function and F(s) the corresponding formal Dirichlet series. The Dirichlet convolution
corresponds to
We therefore have
- ,
giving the Möbius inversion formula
A useful special case is the formula
Mertens conjecture
The Mertens conjecture is that the summatory function
The truth of the Mertens conjecture would imply the Riemann hypothesis. However, computations by Andrew Odlyzko have shown that the Mertens conjecture is false.