Proof
(related to Proposition: Sum of Möbius Function Over Divisors)
- Let n\ge 1 be a natural number.
-
For n=1, the sum of the Möbius function over the divisors of n equals \sum_{d\mid 1}\mu(d)=\mu(1)=1, as required.
-
For n > 1 with the factorization n=p_1^{e_1}\cdots p_r^{e_r}, we have \mu(d)=0 for non square-free divisors d\mid n.
- In the remaining sum, there are exactly \binom rk square-free divisors d with exactly k different prime factors.
- By the definition of the Möbius function, it follows
\begin{array}{rcl}
\sum_{d\mid n}\mu(d)&=&\sum_{d\mid n,\text{ square-free}}\mu(d)=1+\binom{r}{1}(-1)+\binom{r}{2}+\cdots+\binom{r}{2}(-1)^r\\
&=&\sum_{k=0}^r\binom rk(-1)^k.
\end{array}
- The last sum equals by the binomial theorem (1-1)^r=0.
∎
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References
Bibliography
- Scheid Harald: "Zahlentheorie", Spektrum Akademischer Verlag, 2003, 3rd Edition
- Landau, Edmund: "Vorlesungen über Zahlentheorie, Aus der Elementaren Zahlentheorie", S. Hirzel, Leipzig, 1927