I was reading a book called "Math Talks for Undergraduates" by Serge Lang. I was introduced to the Prime Number Theorem that states that $pi(x)$ (representing the number of primes $<=$ x) is asymptotic to $\frac{x}{log(x)}$ using Gauss conjecture that says that the probability of n to be prime is equal to $\frac{1}{log(n)}$.
Then in the second section, I was informed that the proof of the existence of an infinite number of twin primes is still unknown. This is also the case of the prime numbers of the form $k^2+1$.
However, Lang tried to give arguments that not only allow us to guess that there are infinitely many prime numbers of that form, but also to guess how many.
Starting with the twin primes, the probability that n, n+2 are twin primes is equal to $\frac{1}{log^2(n)}$, therefore $pi(x)$ (of twin primes) is asymptotic to $C(\frac{x}{log^2(x)})$ or better it is asymptotic to $C\int{\frac{1}{log^2(t)} dt}$ between 2 and x; where C is a positive constant.
On the other hand, in the case when n is of the form $k^2+1$, Lang used the same process to conclude that $pi(x)$ (of n of the form $k^2+1$) is asymptotic to $C(\frac{\sqrt{x}}{log(x)})$, where C is another positive constant.
I didn't understand how he concluded that the probability that n is of the form $k^2+1$, and is a prime number is equal to $\frac{1}{\sqrt(n) log(n)}$?
Did he use conditional probability? How?
The probability that a number $n$ is prime is $\frac 1{\log n}$. And there are $\sqrt n$ numbers less than or equal to $n$ of the form $k^2+1$ so the probability that $n$ has that form is $\frac 1{\sqrt n}$. It follows that, if these were independent events, you'd get the probability that $n$ had both properties by multiplying these expressions together. That gives Lang's form.
To be clear, there is no particular reason to assume that these are independent events, so this is just a heuristic argument.