  We show a smoothed version of Goldston-Pintz-Yildirim's sifting argument to
detect small gaps between primes, which has a higly flexible error term. Our
argument is applicable to high dimensional Selberg sieve situations as well,
although the relevant details are not stated explicitly.
  In the present v.2, we have added Appendix in which we made a minor
correction to our reasoning following formula (5.14). We are indebted to
Professor Terrence Tao for pointing out the necessity of this amendment. The
text of the original version has not been changed, except for a correction of
the title in the item [2] of the references.
