Hello,

this message follows on from About Goldbach's conjecture and aims at modelling the quantity $N_{2}(n)$ by a random variable.

Let $n$ be a fixed natural number and let's define for every $\varepsilon\gt 0$ the quantity $Y_{k,\varepsilon}$ as the Bernoulli random variable equal to $1$ with probability $p=N_{1}(n)/P_{ord_{C}(n)}$ if $k\lt n$ and equal to $1$ with probability $\varepsilon/k^{2}$ if $k\ge n$. Assume that for any given $\varepsilon$ all of the quantities $Y_{k,\varepsilon}$ are independent. Let $S_{m,\varepsilon}$ be the random variable $\sum_{k=1}^{m}Y_{k,\varepsilon}$. I have been told that $S_{m,\varepsilon}$ converges almost surely towards $X=(S+T(\varepsilon))/np$, where $S$ is the sum of the $n-1$ first terms of the previous sum and $\displaystyle{T(\varepsilon)=\sum_{k=n}^{\infty}Y_{k,\varepsilon}}$, and that the mean value of this random variable $X$ is $1$ as $\varepsilon$ tends to $0$.

So my question is : does this entail that the limit, if it exists, as $n$ goes to infinity of the sequence $\left(\dfrac{N_{2}(n)P_{ord_{C}(n)}}{n.N_1(n)}\right)_{n\in\mathbb{N}}$ is equal to $1$?

Thank you in advance.