Liouville's Function

Liouville's function denotedis defined as follows:

Liouville's function is completely multiplicative since ifandthen

Theorem

Forwe have

Also for all

Proof: Letthenis multiplicative so to determinewe only need to computefor allWe have

Hence ifwe haveIf any exponentis odd thenso If all the exponents are even thenfor allandThis shows thatis a square andotherwise. Also