I am studying bounds in prime gaps and I would like to gather as much information as I could. I am just an undergraduate student, it's not a very important project, I am just doing it by curiosity.
I would like to make the following question (let $p_1,...,p_j$ be the first $j$ primes):
- Let $g_k(n)=\#\{p_{i+1} \,\text{prime} \colon p_i < n, p_{i+1}-p_i \geq k\}$. Are there any significant facts known about this function? Is there any asymptotic formula for this? In particular, is it known if this is asymptotically bigger than $n$?
Thank you, sorry if I made stupid questions.
For any fixed $k$, your function $g_k(n)$ is asymptotic to $\pi(n)$, the number of primes less than $n$; in particular, it is asymptotic to $n/\log n$. In other words, almost all primes pass the test required to be counted by $g_k(n)$.
Sieve methods can establish the following inequality: for any fixed gap $g$, the number of primes $p$ less than $n$ for which $p+g$ is also prime is less than $C(g) n/(\log n)^2$ (when $n$ is sufficiently large). Summing this inequality over all integers $g<k$, we see that the number of primes $p_i$ less than $n$ for which $p_{i+1}<p_i+k$ is at most $D(k) n/(\log n)^2$ for some constant $D(k) = \sum_{g<k} C(g)$. Therefore $\pi(n-1) - D(k) n/(\log n)^2 \le g_k(n) \le \pi(n)$, which is enough to show that $g_k(n) \sim \pi(n)$.