I've been reading Scott Contini's thesis that implements the quadratic sieve.
In the MPQS, Figure 2.2 Trial Division Stage, why is the target value $\log(M\sqrt{N})$ minus a small error term? Shouldn't the target value depend on $x$? I'm worried that, when the absolute value of $x$ is small, a smooth $g(x)$ will be missed because the sum of the logarithms is less than the target.
I understand that the essence of the sieving procedure is to trade a little slop for speed, but small values of $x$ are exactly the place where smooth $g(x)$ are most likely to be found, and that seems like a bad place to accept some slop.