Distribution of primes in smooth moduli

A key input to Zhang's proof that bounded gaps occur infinitely often is a distribution result on primes in smooth moduli, which we have called [math]\displaystyle{ MPZ[\varpi,\delta] }[/math] (and later strengthened to [math]\displaystyle{ MPZ'[\varpi,\delta] }[/math]. These estimates are obtained as a combination of three other estimates, which we will call [math]\displaystyle{ Type_I[\varpi,\delta,\sigma] }[/math], [math]\displaystyle{ Type_{II}[\varpi,\delta,\sigma] }[/math], and [math]\displaystyle{ Type_{III}[\varpi,\delta,\sigma] }[/math].