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

Definition of estimates

Coefficient sequences

Congruence class systems

Smooth and densely divisible numbers

MPZ

Type I

Type II

Type III

The combinatorial lemma

Type I estimates

Level 1

Level 2

Level 3

Level 4

Level 5

Type II estimates

Level 1

Level 2

Level 3

Level 4

Level 5

Type III estimates

Level 1

Level 2

Level 3

Level 4

Combinations