How dense are primes congruent to 1 and 3 (mod 4)?

Question

There are infinitely many primes of the form $4n+1$ and $4n+3$. In a given interval $[0,N]$ for a large enough $N$ do we expect to see the same number of primes congruent to $1$ and $3$ (mod 4)?

Answer 1

Dirichlet theorem of primes in arithmetic progression says that the asymptotic density of primes of the form $4k+1$ and $4k+3$ are both equal to $\frac{x}{2\log x}$. However in the small scale we observe a phenomenon called Chebyshev bias where in the actual number of primes of the form $4k+3$ are slightly more than those of the form $4k+1$. The first violation of this bias occurs only at $x = 26861$.