What fraction of primes are $3 (mod 4)$?

Question

As $n\rightarrow\infty$, we know that the fraction of primes that are $2 \hspace{1mm}(mod\hspace{1mm}4)$ tends to zero as two is the only prime with that property.

While I am aware that there are infinitely many primes of the form $4k+1$ and infinitely many of the form $4k+3$ what is the distribution like? As $n\rightarrow \infty$ do the number of $4k+1$ primes and $4k+3$ primes grow at different rates?

Answer 1

The Prime Number Theorem for arithmetic progressions states that $$\pi(x;a,n)\sim\frac1{\phi(n)}\frac{x}{\ln x}$$ as $x\to\infty$. Here $a$ is coprime to $n$, and $\pi(x;a,n)$ is the number of primes $p$ with $p\le x$ and $p\equiv a\pmod n$. Also $\phi$ denote's Euler's totient. In particular, $$\pi(x;1,4)\sim\pi(x;3,4)\sim\frac12\frac{x}{\ln x}\sim\frac{\pi(x)}2.$$

But there are more subtle aspects to this. Although $\pi(x;1,4)-\pi(x;3,4)$ is $o(x/\ln x)$ it appears to be more often negative than positive; this is Chebsyshev's bias.

Answer 2

No answer, but a nice reference
There is a quite entertaining article by A.Granville and G. Martin, titled "prime numbers race" in the 2006-volume of MAA's series.

It starts:

INTRODUCTION. There's nothing quite like a day at the races... The quickening of the pulse as the starter's pistol sounds, the thrill when your favorite contestant speeds out into the lead (or the distress if another contestant dashes out ahead of yours), and the accompanying fear (or hope) that the leader might change. And what if the race is a marathon? Maybe one of the contestants will be far stronger than the others, taking the lead and running at the head of the pack for the whole race. Or perhaps the race will be more dramatic, with the lead changing again and again for as long as one cares to watch.

Our race involves the odd prime numbers, separated into two teams depending on the remainder when they are divided by 4:
(table)

In this Mod 4 Race, Team 3 contains the primes of the form 4n+3, and Team 1
contains the primes of the form 4n + 1.
The Mod 4 Race has just two contestants and is quite some marathon because it goes on forever! From the data (...)

(enhancement by me)

Theorem (J.E. Littlewood, 1914). There are arbitrarily large values of x for which there are more primes of the form 4n + 1 up to x than primes of the form 4n + 3. In fact, there are arbitrarily large values of x for which (...)

See: "The American Mathematical Monthly, Vol. 113, No. 1 (Jan., 2006), pp. 1-33"
It's on jstor