[Verse 1]
When you sum up all the pieces, divisors in a row
f of n equals sigma, let the total overflow
But what if you could turn it back, reverse what you have done
Find the hidden summands when the adding race is run
[Chorus]
Möbius inversion, it's the undo button's call
When you know the sum of parts, you can find them one and all
Mu of d times f of n over d, that's the key
Number theory's magic trick, sets the summands free
[Verse 2]
Start with g of d summed over every divisor d
That gives you f of n, it's a transformation spree
But Möbius turns tables with his function mu so bright
Takes the total back apart, brings the pieces to light
[Chorus]
Möbius inversion, it's the undo button's call
When you know the sum of parts, you can find them one and all
Mu of d times f of n over d, that's the key
Number theory's magic trick, sets the summands free
[Bridge]
Here's the secret that makes it work so clean
Sigma mu of d equals one when n is one pristine
But when n is greater, sigma mu gives zero neat
Orthogonality relation makes the proof complete
[Verse 3]
Like calculus has integration paired with its derivative
Möbius gives number theory something reconstructive
Two formulas equivalent, both will do the deed
Mu times f of quotient or mu quotient times f indeed
[Chorus]
Möbius inversion, it's the undo button's call
When you know the sum of parts, you can find them one and all
Mu of d times f of n over d, that's the key
Number theory's magic trick, sets the summands free
[Outro]
So remember when you're summing over divisors in your way
Möbius inversion can reverse it any day
The function's just a sieve that filters out what you need
Mathematical undo makes the hidden summands feed