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I can show that $N(\epsilon)$ is equal to $\epsilon^{-2}$ up to a log factor on each side. The strategy I'll use is to give an upper bound for $\pi(1/2+\epsilon,n)$. Optimizing it, we obtain an upper …

Let $X$ be the total number of plays and let $Y$ be the number of plays for the track with the most plays. I think a good strategy is to estimate $$E [ Y- X | X = X_0 ]$$ It seems like this invol …

Let me finish Qiaochu's answer. Observe that the ratio $\mathbb P ( S_n=n+k) / \mathbb P(S_n=n)$ is close to $1$: It is $$\prod_{j=1}^k \frac{n}{n+j}$$. Using $$ e^{-j/n} \leq \frac{n}{n+j} \leq …

We can make a conjecture based on the function field model. Replace $\mathbb Q$ with $\mathbb F_q(t)$, that is, the function field of $C= \mathbb P^1$. Then any element of $\overline{\mathbb F_q(t)}$ …

This is a heuristic, suggesting $f(n)=O( 1/n^2)$. Consider the sphere of radius $r$. Each cylinder intersects this sphere in a great circle. The great circles divide the sphere into a number of regi …

A reasonable conjecture is that $$ \operatorname{Germain}(x) = 2 C \int_2^x \frac{dy}{\log^2 y} + O( x^{1-\delta} )$$ for some $\delta >0$. This is the Hardy-Littlewood conjecture with power-saving er …

Doing this by the moment method would require understanding the expected number of primes in the interval, the expected number of pairs of primes, triples, etc. and I don't think we have asymptotics for …

Using Burnside's lemma it's not too hard to see that $a(n)$ is asymptotic to $2^{2^n}/|GL_n(\mathbb F_2)|$, since every non-identity element of $GL_n(\mathbb F_2)$ contributes at most $2^{ (3/4) 2^{n …