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I was reading "The Prime Numbers and Their Distribution" by Gérald Tenenbaum and Michel Mendès France, the section about Cramér's Model, and I couldn't prove a couple of results. I would like to start with an introduction of the topic.

Introduction:

Cramér's Model consists in considering a sequence of independent random variables $\{X_n\}_{n=2}^\infty$ which take values 0 and 1, such that $\Bbb P (X_n=1)=1/\log n$ for $n \ge 3$. The definition of $\Bbb P (X_2=1)$ is considered irrelevant for the purposes of the model.

The idea of this is to give a probabilistic model for the behaviour of prime numbers: according to the prime numbers theorem, the number of primes less or equal than $n$, with $n$ sufficiently large, is approximately $n/\log n$, so the "probability that $n$ is prime" would be $1/\log n$.

Now we define $S:=\{n\ge2:X_n=1\}$. The usual way of working with this model seems to be, proving that some property holds almost surely for $S$, and then conjecturing that property holds for $\mathcal P$, the sequence of prime numbers.

Question:

By analogy to the usual notation, we define $\pi_S(x):=\sum_{n\le x} X_n$ (note that $\pi_\mathcal P(x)=\pi(x)$). We also define: $$\upsilon_S(x):=\frac{\pi_S(x)-\mathrm {li}(x)}{\sqrt{2x\frac{\log\log x}{\log x}}}$$ The book states that it's easy to show that $\upsilon_S(x)$ oscillates asymptotically between -1 and 1, but a paper by Cramér only states that it's shown in another paper (which I couldn't find) that $\limsup_{x\to \infty}|\upsilon_S(x)|=1$ holds almost surely. I understand that the statement of the book is that both, $\limsup_{x\to \infty}\upsilon_S(x)=1$ and $\liminf_{x\to \infty}\upsilon_S(x)=-1$, hold almost surely, but I could be wrong since I haven't found the definition of "asymptotic oscillation".

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    $\begingroup$ You might have more success asking only one question per question... you can link them to each other to avoid typing out the preamble each time, but as it is this question is kind of intimidating. $\endgroup$ – Ben Millwood Mar 16 '14 at 14:19
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    $\begingroup$ I believe your interpretation of "asymptotic oscillation" is correct. Look in probability textbooks/monographs for the Law of the Iterated Logarithm; that's the tool that should prove this statement for $v_S(x)$. $\endgroup$ – Greg Martin Aug 1 '14 at 18:09

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