Sum of Möbius Function over Divisors/Examples/210

Example of Sum of Möbius Function over Divisors

$\ds \sum_{d \mathop \divides 210} \map \mu d = 0$

Proof

\(\ds \sum_{d \mathop \divides 210} \map \mu d\)

\(=\)

\(\ds \map \mu 1 + \paren {\map \mu 2 + \map \mu 3 + \map \mu 5 + \map \mu 7} + \paren {\map \mu {2 \times 3} + \map \mu {2 \times 5} + \map \mu {2 \times 7} + \map \mu {3 \times 5} + \map \mu {3 \times 7} + + \map \mu {5 \times 7} } + \paren {\map \mu {2 \times 3 \times 5} + \map \mu {2 \times 3 \times 7} + \map \mu {2 \times 5 \times 7} + \map \mu {3 \times 5 \times 7} } + \map \mu {2 \times 3 \times 5 \times 7}\)

\(\ds \)

\(=\)

\(\ds \dbinom 4 0 + \dbinom 4 1 \paren {-1} + \dbinom 4 2 \paren {-1}^2 + \dbinom 4 3 \paren {-1}^3 + \dbinom 4 4 \paren {-1}^4\)

\(\ds \)

\(=\)

\(\ds 1 - 4 + 6 - 4 + 1\)

\(\ds \)

\(=\)

\(\ds 0\)

$\blacksquare$