## Divisor Functions

For any real or complexand any integerwe define

\[\sigma_{\alpha}(n)= \sum_{d | n}d^{\alpha}\]

is the sum of thepowers of

The functionsare called divisor functions. They are multiplicative becausethe Dirichlet product of two multiplicative functions, but not completely multiplicative.

Whenis the number of divisors of

Whenis the sum of the divisors of

Whenis the sum of the reciprocals of the divisors of

Sinceis multiplicative we have

To computenote that the divisors ofare hence

Becauseis multiplicative (but not completely multiplicative) we can also write

Ifthen

The Dirichlet inverse ofcan also by expressed as a linear combination of thepowers of the divisors of

Theorem

Forwe have

Proof: Sinceandis completely multiplicative we have