Primes in Congruence Classes

Question

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...

Answer 1

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 $x$.

Answer 2

The limit in your question is equal to 1, but it sounds like you're thinking of Chebyshev's bias (Wikipedia); see also this section on the prime counting function.