Dirichlet Inverse

Number Theory Level 2

If $f(n)$ is the Dirichlet inverse of $I_k(n) = n^k$ , what is $|f(n)|?$

Notation: $\mu$ denotes the Möbius function .

n^k

\mu

| \mu |

n^k | \mu |

1 solution

Otto Bretscher
Jan 21, 2016

$I_k*(I_k\mu)=I_k(1*\mu)=I_k\epsilon=\epsilon$ . In the first step we use the fact that $I_k$ is totally multiplicative.