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)?
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see this question and suggested reading – J. W. Tanner Aug 09 '19 at 03:04
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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 $x = 26861$.
Nilotpal Sinha
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Chebyshev bias is about the expected sign of $\sum_{p\le x} (-1)^{(p-1)/2}$ which oscillates but is more often $-$ than $+$ – reuns Aug 09 '19 at 05:30
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This is a fine answer, particularly when you bring up Chebyshev bias. Yet, it should not surprise you that Dirichlet's result on asymptotic densities of primes in various residue classes has been already adequately covered on our site. So, as you gain experience in using the site, you should begin to search more. Finding a near duplicate is often not unduly taxing. If you find a close enough duplicate, then vote to close the new version as such. And post fine additions as answers to the original. BTW posting a new answer also bumps the thread, so that people will see your post! – Jyrki Lahtonen Aug 09 '19 at 06:04