Reciprocal-prime sum in residue class one modulo four
= Reciprocal-prime sum in residue class one modulo four
The progression version of the <Mertens second theorem> gives $\sum_{\substack{p\leq y\\p\equiv1\pmod4}}1/p=\tfrac12\log\log y+c+O(1/\log y)$. Subtracting from the all-<prime> sum shows that the forbidden <primes> $p\equiv3\pmod4$ contribute half the logarithmic density. This is the source of the square root in a <half-dimensional interval sieve>.