0

The Cramér random model for the primes is a random subset ${{\mathcal P}}$ of the natural numbers with ${1 \not \in {\mathcal P}}, {2 \in {\mathcal P}}$, and the events ${n \in {\mathcal P}}$ for ${n=3,4,\dots}$ being jointly independent with ${{\bf P}(n \in {\mathcal P}) = \frac{1}{\log n}}$ (the restriction to ${n \geq 3}$ is to ensure that ${\frac{1}{\log n}}$ is less than ${1}$). Prove that almost surely, the quantity $\frac{1}{x/\log x} |\{n \leq x: n \in {\mathcal P}\}|$ converges to one as ${x \rightarrow \infty}$.

Question: We are supposed to show that the event $\lim_{x \to \infty} \frac{1}{x/\log x} |\{n \leq x: n \in {\mathcal P} \}| = 1$ has probability one. What will be the probability space $\Omega$ used to model such events, and how should one proceeds?

shark
  • 789

1 Answers1

0

Use an infinite product space $\Omega = \prod_{n\geq 3} \{0,1\}$ where $\{0,1\}$ in the $n$th component has the probability measure $\mu_n$ such that $\mu_n(\{1\}) = 1/\log n$ and $\mu_n(\{0\}) = 1-1/\log n$. Give $\Omega$ the product measure.

KCd
  • 46,062
  • As I understand, the only possible event space in this setting will be the discrete algebra of $\Omega$, whose elements specify all possible subsets of the primes according to the Cramér’s model. May you provide some hints on how to solve the original problem with this model? – shark Mar 16 '24 at 03:35
  • By "discrete algebra" do you mean the $\sigma$-algebra of all subsets of $\Omega$? That is not how the product $\sigma$-algebra on $\Omega$ is defined. You should use the $\sigma$-algebra generated by "cylinder sets" in $\Omega$; see treatments of product measures on products of infinitely many probability spaces. This is like the distinction on a product of topological spaces between the product topology (very useful) and the box topology (largely useless). See https://math.stackexchange.com/questions/1370091/definition-of-the-product-sigma-algebra. – KCd Mar 16 '24 at 03:53
  • If you have more questions, talk with the person who assigned you the problem that you were asked to show. – KCd Mar 16 '24 at 03:59
  • Right, one puts the discrete $\sigma$- algebra on ${0,1}$, and forming the product $\sigma$-algebra by using the cylinder sets, or equivalently the elementary sets (the measure-theoretic analogue of cylinder sets in point set topology) as generators. – shark Mar 16 '24 at 04:37
  • What I’m unsure about is the way to control the random variables $\frac{x }{\log x }|{n \leq x: n \in {\mathcal P}}|$, which are indexed by an uncountable set. – shark Mar 16 '24 at 04:45
  • Let $X_n \colon \Omega \to {0,1}$ be projection from the $n$th coordinate and look at the random variable $\sum_{n \leq x} X_n$. There's no need to be bothered by an uncountable set of values $x$ since you can just as well let $x$ run over integers: $n \leq x$ is the same as $n \leq \lfloor x\rfloor$ and $x/\log x \sim N/\log N$ when $N = \lfloor x\rfloor$. – KCd Mar 16 '24 at 05:12
  • By SLLN, we have ${\bf P}(\lim_{x \to \infty} \frac{\sum_{n \leq x} X_n \log n}{\lfloor x \rfloor} = 1) = 1$. However one needs to show that ${\bf P}(\lim_{x \to \infty} \frac{\sum_{n \leq x} X_n \log x}{x} = 1) = 1$, I'm not sure how to proceed due to this slight modification. – shark Mar 18 '24 at 19:04
  • Have you tried to express $\sum_{n =3}^N X_n\log n$ in terms of $(\sum_{n=3}^N X_n)\log N$ plus an extra term using partial summation? – KCd Mar 18 '24 at 22:27
  • The second expression has an extra contribution of $(\log N - \log n)X_n = X_n \log \frac{N}{n}$ for each summand, so $\sum_{n=3}^N X_n \log N = \sum_{n=3}^N X_n \log n + \sum_{n = 3}^N X_n \log \frac{N}{n}$. Is this the way you intended to relate the two terms? – shark Mar 18 '24 at 23:46
  • No. I mean partial summation, a.k.a. summation by parts (a common technique in analytic number theory: look it up) which lets you rewrite $\sum_{n = 3}^N X_n\log n$ in terms of the sums $S_n = X_1 + \cdots + X_n$, which is what you want since the expression you care about has numerator $(X_1 + .... + X_x)\log x$. Since this whole post came about because you are "supposed to show" something, have you tried talking with the course instructor? – KCd Mar 19 '24 at 01:42
  • Thanks for all the feedback. For this Exercise however, I don’t think the author assumes familiarity of basic techniques like partial summation in analytic number theory from the audience. In particular the LLN and the moment methods should be suffice. I’ll consult with the instructor for further details. – shark Mar 19 '24 at 03:04