Added mobius inversion (kth-competitive-programming#169)

* added mobius inversion

* Added floor formula

content/number-theory/chapter.tex

@@ -48,3 +48,17 @@ \section{Estimates}
 	$\sum_{d|n} d = O(n \log \log n)$.
 
 	The number of divisors of $n$ is at most around 100 for $n < 5e4$, 500 for $n < 1e7$, 2000 for $n < 1e10$, 200\,000 for $n < 1e19$.
+
+\section{Mobius Function}
+\[
+	\mu(n) = \begin{cases} 0 & n \textrm{ is not square free}\\ 1 & n \textrm{ has even number of prime factors}\\ -1 & n \textrm{ has odd number of prime factors}\\\end{cases}
+\]
+  Mobius Inversion:
+  \[ g(n) = \sum_{d|n} f(d) \Leftrightarrow f(n) = \sum_{d|n} \mu(d)g(n/d) \]
+  Other useful formulas/forms:
+
+  $ \sum_{d | n} \mu(d) = [ n = 1] $ (very useful)
+
+  $ g(n) = \sum_{n|d} f(d) \Leftrightarrow f(n) = \sum_{n|d} \mu(d/n)g(d)$
+
+ $ g(n) = \sum_{1 \leq m \leq n} f(\left\lfloor\frac{n}{m}\right \rfloor ) \Leftrightarrow f(n) = \sum_{1\leq m\leq n} \mu(m)g(\left\lfloor\frac{n}{m}\right\rfloor)$