  Assuming that there exist (infinitely many) Siegel zeros, we show that the
(Rosser-)Jurkat-Richert bounds in the linear sieve cannot be improved, and
similarly look at Iwaniec's lower bound on Jacobsthal's problem, as well as
minor improvements to the Brun-Titchmarsh Theorem. We also deduce an improved
(though conditional) lower bound on the longest gaps between primes, and rework
Cram\'er's heuristic in this situation to show that we would expect gaps around
$x$ that are significantly larger than $(\log x)^2$.
