is there an elementary proof of $\sum\limits_{d|n}\phi(d)=n$?
4512
Is there a direct, elementary proof of $n = \sum_{k|n} \phi(k)$?
90749
Show that $\sum\nolimits_{d|n} \frac{1}{d} = \frac{\sigma (n)}{n}$ for every positive integer $n$.
117837
Formula for the general Cavalieri Sum: $S_n(p)=\sum\limits_{k=1}^{n} k^p\,\,\,n, p\in\mathbb N$
6366
Combinatorial proof of summation of $\sum\limits_{k = 0}^n {n \choose k}^2= {2n \choose n}$
17157
Proving $\sum\limits_{k=1}^{n}{\frac{1}{\sqrt{k}}\ge\sqrt{n}}$ with induction
76137
Too simple proof for convergence of $\sum_n a_n b_n$?
125465
Proving that $\sum \limits_{d^2|n}\ \mu (d)=\mu^2(n)$, where $\mu$ is the Möbius function.
37012
The number of summands $\phi(n)$
33144
Criterion for $\phi(mn)=m\phi(n)$ to be true
36786
Prove $\sum\limits_{i=0}^n\binom{i+k-1}{k-1}=\binom{n+k}{k}$ (a.k.a. Hockey-Stick Identity)
151111
How do I show that $\frac{n}{\varphi(n)}=\sum\limits_{d \mid n} \frac{\mu^2(d)}{\varphi(d)}$?
23557
Algebraic Proof that $\sum\limits_{i=0}^n \binom{n}{i}=2^n$
47790
If $x \geq 2$ prove that $\sum_{ n \leq x} \frac{1}{\phi(n)} = \mathcal{O}(\log(x))$
121723
$\phi(n)=\frac{n}{2}$ if and only if $n=2^k$ for some positive integer k
101017
What is the sum of $\sum\limits_{i=1}^{n}ip^i$?
138819
Prove that $\phi(n) \geq \sqrt{n}/2$
62504
Is there any known relationship between $\sum_{i=1}^{n} f(i)$ and $\sum_{i=1}^{n} \dfrac {1}{f(i)}$