Half-dimensional interval sieve

Sieve an interval of length $z$ by the prime two and the primes congruent to three modulo four, with local densities $g(p)=1/p$. Take $D=z^{1/2}$ and $L=\sqrt{D}$. The truncated Euler-product lower bound, the Mertens first theorem and the reciprocal-prime sum in residue class one modulo four give $J\gg \sqrt{\log z}$. The interval counts have bounded remainders, whose sum is $O(D{\log }^{2}D)$ by the summatory bound for three to the prime omega. Hence the number surviving is $O(z/\sqrt{\log z})$, uniformly in the location of the interval.