Intereting Posts

Finite Element Method for a Two-Point Problem
Does the law of the excluded middle imply the existence of “intangibles”?
What needs to be linear for the problem to be considered linear?
How to reduce higher order linear ODE to a system of first order ODE?
Efficiently partition a set into all possible unique pair combinations
How prove this $|ON|\le \sqrt{a^2+b^2}$
Pacman on a Mobius Strip
Complex Lie algebra $\mathfrak{g}$ is solvable implies that $\mathfrak{g}'$ is nilpotent.
How to know if a point is analytics or not?
In how many different ways can we fully parenthesize the matrix product?
Reduced schemes and global sections
The final state of 1000 light bulbs switched on/off by 1000 people passing by
Find the center of the symmetry group $S_n$.
Closure of open set in a dense subspace of topological space.
A and B is similar ⇒ $A^T$ is similar to $B^T$.

In David Williams’ Probability with Martingales, $\exists$ this exercise.

Let $s > 1$ and let $\zeta(s) = \sum_{n=1}^\infty n^{-s}$.

Let $X$ and $Y$ be independent $\mathbb{N}$-valued random variables with $P(X=n)=P(Y=n)=\dfrac{n^{-s}}{\zeta(s)}$.

- Limits of sequences of sets
- Conditional probability and the disintegration theorem
- Proof of fundamental theorem of integral calculus
- Reconciling several different definitions of Radon measures
- Is it always true that $\sum^{\infty}a_{i}1_{A_{i}}-\sum^{\infty}b_{i}1_{B_{i}}=\sum^{\infty}c_{i}1_{C_{i}}$?
- Bartoszyński's results on measure and category and their importance

Facts:

1 The events ($E_p$: $p$ prime) where $E_p =$ ($X$ is divisible by $p$) (This time $p$ may not be prime) are independent.

2 The events ($F_p$: $p$ prime) where $F_p =$ ($Y$ is divisible by $p$) (This time $p$ may not be prime) are independent.

3 $P(E_p) = p^{-s}$

4 This is Euler’s formula $1/\zeta(s) = \prod_p (1-1/p^s)$ where $p$ is prime.

5 $P(X=1) = 1/\zeta(s) = \prod_p (1-1/p^s)$

6 $P(\text{no square other than 1 divides }X)= \dfrac{1}{\zeta(2s)}$

Here is the exercise:

Let $H$ be the highest common factor of $X$ and $Y$. Prove that $P(H=n) = \dfrac{n^{-2s}}{\zeta(2s)}$.

How do I go about doing this? I know $H$ divides both $X$ and $Y$ but that’s just making use of the fact that $H$ is a common factor. Since it is the highest, all the natural numbers greater than $H$ do not divide $X$ and $Y$ or something like that?

- Prove or disprove that $\exists a,b,c\in\mathbb{Z}_+\ \forall n\in\mathbb{Z}_+\ \exists k\in\mathbb{Z}_+\colon\ a^k+b^k+c^k = 0(\mathrm{mod}\ 2^n)$
- The number of partitions of $n$ into distinct parts equals the number of partitions of $n$ into odd parts
- Predicting the number of decimal digits needed to express a rational number
- Is there any perfect squares that are also binomial coefficients?
- Trying to generalize an inequality from Jitsuro Nagura: Does this work?
- Generalizing $(-1)^{n}$ by using $k$th-level figurate numbers
- Tight bounds for Bowers array notation
- What are the bases $\beta$ such that a number with non-periodic expansion can be approximated with infinitely many numbers with periodic expansion?
- For an integer $n \geq 1$, verify the formula: $\sum\limits_{d|n} \mu (d) \lambda(d)= 2^{\omega (n)}$
- Relationship between prime factorizations of $n$ and $n+1$?

I don’t really understand what you mean in facts 1 and 2, about $p$ being prime but may not be prime. But anyway:

\begin{align*}

P(H=n) &= \sum_{\substack{c,d\in\Bbb N \\ \gcd(c,d)=n}} P(X=c)P(Y=d) \\

&= \sum_{\substack{a,b\in\Bbb N \\ \gcd(a,b)=1}} P(X=na)P(Y=nb) \\

&= \sum_{\substack{a,b\in\Bbb N \\ \gcd(a,b)=1}} \frac{(na)^{-s}}{\zeta(s)}\frac{(nb)^{-s}}{\zeta(s)} \\

&= n^{-2s} \sum_{\substack{a,b\in\Bbb N \\ \gcd(a,b)=1}} \frac{a^{-s}}{\zeta(s)}\frac{b^{-s}}{\zeta(s)} \\

&= n^{-2s} \sum_{\substack{a,b\in\Bbb N \\ \gcd(a,b)=1}} P(X=a)P(Y=b) = n^{-2s} P(H=1).

\end{align*}

Therefore

$$

1 = \sum_{n=1}^\infty P(H=n) = \sum_{n=1}^\infty n^{-2s} P(H=1) = \zeta(2s)P(H=1),

$$

and the desired result follows.

- Image of the union and intersection of sets.
- Solution of the integral equation $y(x)+\int_{0}^{x}(x-s)y(s)ds=x^3/6$
- Convergence/Divergence of infinite product
- Let $Z$ be standard normal can we find a pdf of $(Z_1,Z_2)$ where $Z_1=Z \cdot 1_{S},Z_2=Z \cdot 1_{S^c}$
- What is the meaning of evaluating the divergence at a _point_?
- How to prove $b=c$ if $ab=ac$ (cancellation law in groups)?
- eqiuvalent norms in $H_0^2$
- Interchanging Summation and Integral
- Algebraic Proof that $\sum\limits_{i=0}^n \binom{n}{i}=2^n$
- Proving that these two sets are denumerable.
- $\sum_{p \in \mathcal P} \frac1{p\ln p}$ converges or diverges?
- Geometry Book Recommendation?
- Differentiability implying continuity
- A problem on positive semi-definite quadratic forms/matrices
- Vector Algebra Coordinate Transformation