## Mobius function “recursion”

Update 18.2.2017:

The recurrence for the Möbius function is:

$\mu(n) = -{\sum_{d|n}}_{d

The latex code should have been:
latex \mu(n) = -{\sum_{d|n}}_\limits{d<n} \mu(d)
but WordPress does not parse the limits command.

http://math.stackexchange.com/a/853300/8530

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Someone searched the Internet for "mobius function recursion" and found this blog so this post will be about how to view the Mobius function in terms of recursion, although the mobius function “mu” doesn’t have a recurrence/recursion.  (I am not sure any more. I read in Wikipedia that it has a recursion.) Anyhow here follows examples of what the “recursion” looks like:

mu(1) = 1

mu(2)=-mu(1)=-1

mu(3)=-mu(1)=-1

mu(4)=-mu(1)-mu(2)=-1-(-1)=-1+1=0

mu(5)=-mu(1)=-1

mu(6)=-mu(1)-mu(2)-mu(3)=-1-(-1)-(-1)=-1+1+1=1

mu(7)=-mu(1)=-1

mu(8)=-mu(1)-mu(2)-mu(4)=-1-(-1)-0=-1+1-0=0

mu(9)=-mu(1)-mu(3)=-1-(-1)=-1+1=0

mu(10)=-mu(1)-mu(2)-mu(5)=-1-(-1)-(-1)=-1+1+1=1

mu(11)=-mu(1)=-1

mu(12)=-mu(1)-mu(2)-mu(3)-mu(4)-mu(6)=-1-(-1)-(-1)-0-(1)=-1+1+1-0-1=0

mu(13)=-mu(1)=-1

mu(14)=-mu(1)-mu(2)-mu(7)=-1-(-1)-(-1)=-1+1+1=1

mu(15)=-mu(1)-mu(3)-mu(5)=-1-(-1)-(-1)=-1+1+1=1

mu(16)=-mu(1)-mu(2)-mu(4)-mu(8)=-1-(-1)-0-0=-1+1-0-0=0

mu(17)=-mu(1)=-1

mu(18)=-mu(1)-mu(2)-mu(3)-mu(6)-mu(9)=-1-(-1)-(-1)-(1)-0=-1+1+1-1-0=0

mu(19)=-mu(1)=-1

mu(20)=-mu(1)-mu(2)-mu(5)-mu(10)=-1-(-1)-(-1)-(1)=-1+1+1-1=0

mu(21)=-mu(1)-mu(3)-mu(7)=-1-(-1)-(-1)=-1+1+1=1

mu(22)=-mu(1)-mu(2)-mu(11)=-1-(-1)-(-1)=-1+1+1=1

mu(23)=-mu(1)=-1

mu(24)=-mu(1)-mu(2)-mu(3)-mu(4)-mu(6)-mu(8)-mu(12)=-1-(-1)-(-1)-0-(1)-0-0=-1+1+1-0-1-0-0=0

mu(25)=-mu(1)-mu(5)=-1-(-1)=-1+1=0

mu(26)=-mu(1)-mu(2)-mu(13)=-1-(-1)-(-1)=-1+1+1=1

mu(27)=-mu(1)-mu(3)-mu(9)=-1-(-1)-0=-1+1-0=0

mu(28)=-mu(1)-mu(2)-mu(4)-mu(7)-mu(14)=-1-(-1)-0-(-1)-(1)=-1+1-0+1-1=0

mu(29)=-mu(1)=-1

mu(30)=-mu(1)-mu(2)-mu(3)-mu(5)-mu(6)-mu(10)-mu(15)=-1-(-1)-(-1)-(-1)-(1)-(1)-(1)=-1+1+1+1-1-1-1=-1

A European dot comma style Excel formula for the above:

 =IF(COLUMN()=1;1;IF(ROW()=COLUMN();-SUM(INDIRECT(ADDRESS(ROW();1)&":"&ADDRESS(ROW();COLUMN()-1)));IF(ROW()>COLUMN();INDIRECT(ADDRESS(ROW()-COLUMN();COLUMN()));0))) 

which is a recurrence for the Mobius function.