query: $P(n) = n^{ \phi(n)} \prod_{d|n}{(d!/d^d)}^{\mu(n/d)} $
positive_passages.docid: 144819
positive_passages.text: Proving $P(n) =n^{\phi(n)} \prod\limits_{d \mid n} \left(\frac{d!}{d^d} \right)^{\mu(n/d)}$
negative_passages.docid: 135219
negative_passages.text: On the mean value of a multiplicative function: Prove that $\sum\limits_{n\leq x} \frac{n}{\phi(n)} =O(x) $
negative_passages.docid: 84930
negative_passages.text: How to justify $\phi(n) \ge \sqrt{n}$
negative_passages.docid: 134007
negative_passages.text: Borel-Cantelli-related exercise: Show that $\sum_{n=1}^{\infty} p_n < 1 \implies \prod_{n=1}^{\infty} (1-p_n) \geq 1- S$.
negative_passages.docid: 7448
negative_passages.text: Prove $S \doteq \sum_{n=1}^{\infty} p_n  0$ assuming $0 \leq p_n < 1$.
negative_passages.docid: 92569
negative_passages.text: Showing that ${\rm E}[X]=\sum_{k=0}^\infty P(X>k)$ for a discrete random variable
negative_passages.docid: 58079
negative_passages.text: How does $ \sum_{p<x} p^{-s} $ grow asymptotically for $ \text{Re}(s) < 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: 42366
negative_passages.text: Prove $\mathbb{P}(\sup_{t \geq 0} M_t > x \mid \mathcal{F}_0)= 1 \wedge \frac{M_0}{x}$ for a martingale $(M_t)_{t \geq 0}$
negative_passages.docid: 113612
negative_passages.text: Why does $P(\theta_1\mid D, \theta_2) \propto P(D \mid \theta_1, \theta_2)P(\theta_1)$ hold?
negative_passages.docid: 151111
negative_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: 47405
negative_passages.text: sufficiency and necessity of convergence of $\sum a_n$ wrt convergence of $\prod (1 + a_n)$
negative_passages.docid: 30356
negative_passages.text: $a =\prod_{i = 1}^{r} p_{i}^{ai}$ with $a_{i} > 0$ for each $i$ is the canonical representation of $a$...
negative_passages.docid: 88656
negative_passages.text: For $p$ prime, show that $\Phi_{p^n}(x) = 1 + x^{p^{n-1}} + x^{2p^{n-2}} + \dots +x^{(p-1)^{p^n-1}}$
negative_passages.docid: 94945
negative_passages.text: Sum and product of Martingale processes
negative_passages.docid: 74481
negative_passages.text: Distribution of scalar products of two random unit vectors in $D$ dimensions
negative_passages.docid: 88662
negative_passages.text: $n$ divides $\phi(a^n -1)$ where $a, n$ are positive integer.