# Distribution of primes in smooth moduli

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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 $MPZ[\varpi,\delta]$ (and later strengthened to $MPZ'[\varpi,\delta]$. These estimates are obtained as a combination of three other estimates, which we will call $Type_I[\varpi,\delta,\sigma]$, $Type_{II}[\varpi,\delta,\sigma]$, and $Type_{III}[\varpi,\delta,\sigma]$.