Q:

Primes in Congruence Classes

Let $\pi_1(x)$ denote the number of primes $p$ less than $x$ such that $p \equiv 1 \bmod 4$ and $\pi_3(x)$ the number of primes $p$ less than $x$ such that $p \equiv 3 \bmod 4$. How does one approach $$\lim_{x\to \infty}\frac{\pi_1(x)}{\pi_3(x)}$$ I remember seeing a theorem addressing this problem, but cannot remember its name...

A:

Its the combination of the Prime Number Theorem and Dirichlet's Theorem
on primes in arithmetic progressions.  This states that
$$\pi_{a,n}(x)\sim\frac1{\phi(n)}\frac{x}{\ln x}$$
where $a$ is coprime to $n$ and $\pi_{a,n}(x)$ is the number of primes
$\equiv a\pmod n$ up to $x$.

