[Verse 1]
When functions dance with products clean
There's magic hiding in between
If m and n share no common thread
Then f of mn splits like bread
[Chorus]
Multiplicative, break it down
When coprime numbers come around
f of mn equals f of m times f of n
That's the rule that lets us win
Tau and sigma, phi and mu
All obey this golden clue
[Verse 2]
Start with primes raised to a power
That's where multiplicative shows its hour
Tau counts divisors, simple and neat
p to the a gives a plus one complete
[Chorus]
Multiplicative, break it down
When coprime numbers come around
f of mn equals f of m times f of n
That's the rule that lets us win
Tau and sigma, phi and mu
All obey this golden clue
[Bridge]
Sigma sums with geometric grace
p to the a plus one minus one divided by p minus one in place
Phi counts totients standing proud
p to the a minus p to the a minus one allowed
[Verse 3]
Product notation tells the tale
When prime factorizations never fail
Each exponent plays its part
Multiplicative functions are high art
[Chorus]
Multiplicative, break it down
When coprime numbers come around
f of mn equals f of m times f of n
That's the rule that lets us win
Tau and sigma, phi and mu
All obey this golden clue
[Outro]
GCD of one unlocks the gate
Multiplicative seals our fate