Montgomery's sieve allows a general estimate for the
mean value sum of an arbitrary arithmetic function
over residue classes. There are only some limited
applications of it, such as the number of twin primes,
resp. Goldbach exceptions in APs.
Concerning the latter, we conclude that there are
O(n/q (log n)-A)
many Goldbach exceptions up to n
in any residue class modulo q
for almost all q up to n1/2.
This result may be improved considerably by the new
work of J. Pintz on the number of Goldbach exceptions.