Show $\sum\limits_{d|n}\phi(d) = n$.
4512
Is there a direct, elementary proof of $n = \sum_{k|n} \phi(k)$?
43885
Proving that $\sum\limits_{n = 0}^{2013} a_n z^n \neq 0$ if $a_0 > a_1 > \dots > a_{2013} > 0$ and $|z| \leq 1$
125465
Proving that $\sum \limits_{d^2|n}\ \mu (d)=\mu^2(n)$, where $\mu$ is the Möbius function.
104549
When $d \mid n$, the number of elements of order $d$ in a cyclic subgroup of order $n$ is $\phi (d)$
36786
Prove $\sum\limits_{i=0}^n\binom{i+k-1}{k-1}=\binom{n+k}{k}$ (a.k.a. Hockey-Stick Identity)
84930
How to justify $\phi(n) \ge \sqrt{n}$
14197
show that $\sum_{i=0}^l \dfrac{{l\choose i}}{{n\choose k + i}} = \dfrac{n+1}{(n-l+1) {n-l\choose k}}$
67454
Show that $\sum_{i=1}^n (x_i - \bar{x})(y_i - \bar{y}) = \sum_{i=1}^n (x_i - \bar{x})(y_i)$.
59260
How to prove the convergence of $\sum_{n\geq 0}\frac{n!^d}{(dn)!}$ for $d\geq 2$?
108921
By substituting $z = h(t)$ show that $\delta(h(t))=\sum\limits_{i}\frac{\delta(t−t_i)}{\mid h^{\prime}(t_i)\mid}$
17067
To prove that $(\mathbb{P(N)},d)$ is a metric space
147705
Show that $\mathbb{E}(T) = \sum\limits^\infty_{k=1}\mathbb{P}(T \geq k)$ for $T$ nonnegative integer valued and $E[T] < \infty$
137982
Show that $\textbf{ZF}$ shows $\phi$ is $\Delta_1$
60031
Showing $\sum\limits_{j=0}^M \frac{M \choose j}{N+M \choose j} = \frac{N+M+1}{N+1}$
634
Closed form formula for $\sum\limits_{k=1}^n k^k$
35447
If $\sum a_n$ converges and $b_n=\sum\limits_{k=n}^{\infty}a_k $, prove that $\sum \frac{a_n}{b_n}$ diverges
115258
Find all positive integers $n$ such that $\phi(n)+\sigma(n)=2n$.