Relative prime density of $f(n)$
Definition
Number of primes of $f(n)$
Using the prime counting function $\pi(n)$, we can define the number of primes $\pi_{nbr}$ of a function $f(n)$ from $n=1$ to $k$ as:
$$\pi_{nbr}(f(n),k)=\sum_{n=1}^{k}\pi(f(n))-\sum_{n=1}^{k}\pi(f(n)-1)$$
Expected number of primes of $f(n)$
The prime number theorem states that the probability that a integer $n$ is prime is $\approx\frac{1}{\log n}$, so we can define the expected number of primes $\pi_{exp}$ of a function $f(n)$ from $n=1$ to $k$ as:
$$\pi_{exp}(f(n),k)=\sum_{n=1}^{k}\frac{1}{\log f(n)}$$
Relative prime density of $f(n)$
For a function $f(n)$ from $n=1$ to $k$, if we take the number of primes over the expected number of primes, we get the relative prime density $\rho_{rel}$ of $f(n)$:
$$\rho_{rel}(f(n),k)=\frac{\pi_{nbr}(f(n),k)}{\pi_{exp}(f(n),k)}=\frac{\sum_{n=1}^{k}\pi(f(n))-\sum_{n=1}^{k}\pi(f(n)-1)}{\sum_{n=1}^{k}\frac{1}{\log f(n)}}$$
In other words, if you choose a random integer of the form $f(n)$, not greater than $f(k)$, you have $\rho_{rel}(f(n),k)$ times more chance of finding a prime number than if you choose from any integer not greater than $f(k)$.
Observations
A) $\rho_{rel}$ of the $n^{th}$ prime $p_{n}$
$$\rho_{rel}(p_{n},k)=\frac{\pi_{nbr}(f(n),k)}{\pi_{exp}(f(n),k)}=\frac{k}{\sum_{n=1}^{k}\frac{1}{\log p_{n}}}$$
We know from the PNT that the expected number of primes $\pi_{exp}$ of a function $f(n)$ from $n=1$ to $k$ can also be defined as:
$$\pi_{exp}(f(n),k)\sim\frac{k}{\log f(k)}$$
in the sens that:
$$\lim_{k\rightarrow\infty}=\frac{\frac{k}{\log f(k)}}{\sum_{n=1}^{k}\frac{1}{\log f(n)}}=1$$
Therefore, we get:
$$\rho_{rel}(p_{n},k)=\frac{\pi_{nbr}(f(n),k)}{\pi_{exp}(f(n),k)}\sim\frac{k}{\frac{k}{\log p_{k}}}\sim\log p_{k}$$
And finally: $$\lim_{k\rightarrow\infty}\rho_{rel}(p_{n},k)=\log\infty=\infty$$
$k$ $\rho_{rel}(p_{n},k)$ $\log p_{k}$ $\frac{\log p_{k}}{\rho_{rel}(p_{n},k)}$
2^1 0.850002 1.098612 1.29248
2^2 1.146733 1.945910 1.69691
2^3 1.603961 2.944438 1.83572
2^4 2.216540 3.970291 1.79121
2^5 2.965244 4.875197 1.64411
2^6 3.816873 5.739792 1.50379
2^7 4.726268 6.577861 1.39176
2^8 5.653045 7.389563 1.30718
2^9 6.571521 8.208219 1.24905
2^10 7.467793 9.007121 1.20612
2^11 8.338007 9.790486 1.17419
2^12 9.183391 10.56805 1.15077
2^13 10.00733 11.33877 1.13304
2^14 10.81376 12.10350 1.11926
2^15 11.60615 12.86383 1.10836
2^16 12.38744 13.61905 1.09942
2^17 13.15982 14.37085 1.09202
2^18 13.92501 15.11873 1.08572
2^19 14.68433 15.86372 1.08031
B) $\rho_{rel}$ of $f(n)=xn-1$
$$\rho_{rel}(xn-1,k)=\frac{\pi_{nbr}(xn-1,k)}{\pi_{exp}(xn-1,k)}=\frac{\sum_{n=1}^{k}\pi(xn-1)-\sum_{n=1}^{k}\pi(xn-2)}{\sum_{n=1}^{k}\frac{1}{\log(xn-1)}}$$
When we compute $\rho_{rel}(xn-1,k)$ for different values of $x$ , we can clearly see that the values converge, and that:
$$\lim_{k\rightarrow\infty}\rho_{rel}(xn-1,k)=\prod_{x\equiv0\pmod p}\left(1+\frac{1}{p-1}\right)$$
$x$ $\rho_{rel}(xn-1,10^{5})$ $\prod_{x\equiv0\pmod p}\left(1+\frac{1}{p-1}\right)$
2 1.990876 2
3 1.487079 1.5
4 1.992607 2
5 1.241260 1.25
6 2.987535 3
7 1.161095 1.1666666
8 1.980295 2
9 1.493497 1.5
10 2.494988 2.5
11 1.096479 1.1
12 2.986270 3
13 1.084067 1.083333333
14 2.323696 2.333333333
15 1.863521 1.875
16 1.987482 2
17 1.065104 1.0625
18 2.995015 3
19 1.041543 1.055555556
20 2.483307 2.5
21 1.745753 1.75
22 2.198750 2.2
23 1.046219 1.045454545
24 2.978536 3
25 1.252011 1.25
26 2.152474 2.166666667
27 1.495796 1.5
28 2.330276 2.333333333
29 1.039471 1.035714286
30 3.733861 3.75
Questions
I'm trying to answer the following questions:
1) Show that:
$$\lim_{k\rightarrow\infty}\rho_{rel}(xn-1,k)=\frac{\sum_{n=1}^{k}\pi(xn-1)-\sum_{n=1}^{k}\pi(xn-2)}{\sum_{n=1}^{k}\frac{1}{\log(xn-1)}}=\prod_{x\equiv0\pmod p}\left(1+\frac{1}{p-1}\right)$$
2) Prove/disprove the following conjecture:
The limit $\lim_{k\rightarrow\infty}\rho_{rel}(f(n),k)$ is defined for any functions except $f(n)=p_{n}$ , where $p_{n}$ is the $n^{th}$ prime.
3) Prove/disprove the following conjecture:
There exists no function $f(n)$ such that $\rho_{rel}(f(n),k)\geq\rho_{rel}(p_{n},k)$ for sufficiently large values of $k$ , where $p_{n}$ is the $n^{th}$ prime.
