[Verse 1]
When two numbers share no common ground
Their greatest common divisor is one
A special function can be found
That splits apart what seems like one
If m and n are coprime friends
Then f of m times n transcends
To f of m times f of n
This multiplicative blend
[Chorus]
Multiplicative magic, split and multiply
When g-c-d is one, the function won't lie
f of m-n equals f of m times f of n
Tau and sigma, phi and mu, they follow this trend
Prime powers hold the key, that's where it begins
Multiplicative magic, let the pattern sink in
[Verse 2]
Tau counts divisors, here's the way
For p to the power a
Add one more to a, that's your play
a plus one is the formula
When numbers factor into primes
Each power follows these same rhymes
Multiply each piece you've found
The total count comes safe and sound
[Chorus]
Multiplicative magic, split and multiply
When g-c-d is one, the function won't lie
f of m-n equals f of m times f of n
Tau and sigma, phi and mu, they follow this trend
Prime powers hold the key, that's where it begins
Multiplicative magic, let the pattern sink in
[Bridge]
Sigma sums divisors up
p to a plus one minus one
Divided by p minus one, that's the cup
Phi counts numbers coprime, run
p to a minus p to a minus one
Each prime power gets its turn
Multiply them when you're done
[Verse 3]
Four functions dancing in the light
Tau, sigma, phi, and mu
Each one multiplicative and bright
Prime factorizations guide you through
Break down n to its prime parade
Each power has its value made
Multiply across the chain
The full result you will obtain
[Chorus]
Multiplicative magic, split and multiply
When g-c-d is one, the function won't lie
f of m-n equals f of m times f of n
Tau and sigma, phi and mu, they follow this trend
Prime powers hold the key, that's where it begins
Multiplicative magic, let the pattern sink in
[Outro]
When coprime factors come together
Multiplicative functions know
Split them up like birds of a feather
Watch the beautiful pattern flow