[Verse 1]
Let me tell you bout functions that count and sum
Tau and sigma, here they come
Tau of n counts divisors all
Every factor, big and small
Take twelve, its factors are one two three four six twelve
Tau of twelve equals six, now you can tell
[Chorus]
Tau counts them, sigma sums them up
Möbius flips like a measuring cup
Tau of n, divisors we find
Sigma adds them, peace of mind
Möbius dances, positive negative zero
These functions make you a number theory hero
[Verse 2]
Sigma takes those divisors true
Adds them up for me and you
Twelve again, lets do the math
One plus two plus three plus four plus six plus twelve on this path
Twenty-eight is what we get
Sigma twelve, dont you forget
[Chorus]
Tau counts them, sigma sums them up
Möbius flips like a measuring cup
Tau of n, divisors we find
Sigma adds them, peace of mind
Möbius dances, positive negative zero
These functions make you a number theory hero
[Verse 3]
Now Möbius is special friend
On prime powers it depends
Mu of one equals one, thats where we start
If n has squared prime factors, mu is zero from the heart
But if n has k distinct primes all different and free
Mu equals negative one to the k, thats the key
[Bridge]
Six equals two times three, two primes distinct
Mu of six is positive one, faster than you blink
Twelve has two squared, so mu is zero there
Thirty equals two times three times five, negative one we declare
[Chorus]
Tau counts them, sigma sums them up
Möbius flips like a measuring cup
Tau of n, divisors we find
Sigma adds them, peace of mind
Möbius dances, positive negative zero
These functions make you a number theory hero
[Outro]
Three functions dancing in number theory land
Tau and sigma and mu, now you understand