TITLE: What is the proportion of primes that can be written as $a^2 + b^2$?
QUESTION [5 upvotes]: We know that a prime $p$ can be written as a sum of squares if $p \equiv 1 \pmod 4$.
Do we have any idea what proportion of primes are of the form $4k + 1$ for some $k$?
I suppose it would require more knowledge about prime numbers, though I'm not sure what work has been done on this.

REPLY [2 votes]: A very nice paper talking about all this stuff is  Prime Number Races .
You will find it interesting!