A sharper Type I/II estimate for primes in arithmetic progressions to smooth moduli, and H_m << e^{3.8043 m}

In Stadlmann's Type I/II equidistribution estimate for x^delta-smooth moduli, one of the three conditions, 64 varpi + 20 delta + 2 sigma < 1, can be replaced by 48 varpi + 18 delta + 2 sigma < 1. This weaker condition follows from the second one, 48 varpi + 16 delta + 4 sigma < 1, whenever delta <= sigma. The proof follows Stadlmann's argument and makes two changes: the length of a block in one Taylor expansion depends on the pair of shifts, and the bounds are summed over the pairs instead of being multiplied by their number. Combined with Stadlmann's general Harman-sieve construction, this gives a minorant for the primes with exponent of distribution 1/2 + 2 varpi to smooth moduli for every varpi < 7/544, instead of varpi = 0.01265. As a consequence, liminf (p_{n+m} - p_n) << e^{cm} for every c > 3.80428, and for every c > 3.80423 if a numerically computed value of the sieve deficit is used. This improves Stadlmann's exponent 3.8075.