query: A question about Euler's Totient: $\frac1 {\phi(n)} = \frac1 n \sum_{d|n} \frac{\mu(d)^2}{\phi(d)}$
positive_passages.docid: 151111
positive_passages.text: How do I show that $\frac{n}{\varphi(n)}=\sum\limits_{d \mid n} \frac{\mu^2(d)}{\varphi(d)}$?
negative_passages.docid: 4530
negative_passages.text: For which natural numbers are $\phi(n)=2$?
negative_passages.docid: 880
negative_passages.text: Find all n such that $\phi(n) = n/2$
negative_passages.docid: 3356
negative_passages.text: Why $\sum_{k=0}^m x^k $ is equal to $\dfrac{x^{m+1}-1}{x-1} $?
negative_passages.docid: 145835
negative_passages.text: Closed form for CRT solution using Euler $\phi\,$ (totient)
negative_passages.docid: 158582
negative_passages.text: Prove that $\{\frac{\phi (n)}{n}\}_{n \in \Bbb N}$ is dense in $[0,1]$
negative_passages.docid: 126105
negative_passages.text: Compute $\sum_{k=1}^{n} \frac 1 {k(k + 1)} $
negative_passages.docid: 100515
negative_passages.text: Prove that if $d$ divides $n$ then $φ(d)$ divide $φ(n)$ for $φ$ denotes Euler’s φ-function.
negative_passages.docid: 96039
negative_passages.text: Why is $\frac {1\cdot2\cdot3\cdot...\cdot n}{(n+1)(n+2)...(2n)}\le \frac 1 {n+1}$
negative_passages.docid: 89728
negative_passages.text: Prove that $1<\frac{1}{n+1}+\frac{1}{n+2}+...+\frac{1}{3n+1}$
negative_passages.docid: 139074
negative_passages.text: Question about $a_n=\sum_{k=1}^n \frac{n}{n^2+k}\ $for $ n \in\mathbb{N}$
negative_passages.docid: 102334
negative_passages.text: How can you prove $\frac{n(n+1)(2n+1)}{6}+(n+1)^2= \frac{(n+1)(n+2)(2n+3)}{6}$ without much effort?
negative_passages.docid: 124220
negative_passages.text: Using Euler's Totient Function, how do I find all values n such that $\phi(n)=12$?
negative_passages.docid: 48642
negative_passages.text: Let $f(n) = \sum_{(k,n)=1} e^\frac{2\pi ik}{n}$. Show that $f(n) = \sum_{k=1}^{n}\left(\sum_{d\mid (k,n)} \mu(d)e^{\frac{2\pi ik}{n}} \right) $
negative_passages.docid: 14197
negative_passages.text: show that $\sum_{i=0}^l \dfrac{{l\choose i}}{{n\choose k + i}} = \dfrac{n+1}{(n-l+1) {n-l\choose k}}$
negative_passages.docid: 62382
negative_passages.text: I suppose $(a,m)=(a-1,m)=1$, show that $1+a+a^2+\ldots+a^{\phi(m)-1}\equiv0\pmod m$
negative_passages.docid: 150642
negative_passages.text: How to prove $\sum^n_{i=1} \frac{1}{i(i+1)} = \frac{n}{n+1}$?