Competition problem — Number theory

Library / /248 of 520

Number theory Difficulty 6.1 National olympiad Prove it

18. Let $f$ be an arithmetic function. Show that if $F$ is the arithmetic function defined by
$F(n)=\sum_{d \mid n} f(d)$
then
$f(n)=\sum_{d \mid n} \mu(d) F(n / d)$

This result is called the Möbius inversion formula.

Solution

None

Want a route through all this instead of an archive? The track puts 2,000 problems in a working order, from AMC 10 level to the IMO shortlist.

Source: NuminaMath-1.5, licensed Apache-2.0. Statement and solution reproduced as published; topic and difficulty added by this site.