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].


Asymptotic notation

[math]x[/math] is a parameter going off to infinity, and all quantities may depend on [math]x[/math] unless explicitly declared to be "fixed". The asymptotic notation [math]O(), o(), \ll[/math] is then defined relative to this parameter. A quantity [math]q[/math] is said to be of polynomial size if one has [math]q = O(x^{O(1)})[/math], and bounded if [math]q=O(1)[/math]. We also write [math]X \lessapprox Y[/math] for [math]X \ll x^{o(1)} Y[/math], and [math]X \sim Y[/math] for [math]X \ll Y \ll X[/math].

Coefficient sequences

We need a fixed quantity [math]A_0\gt0[/math].

A coefficient sequence is a finitely supported sequence [math]\alpha: {\mathbf N} \rightarrow {\mathbf R}[/math] that obeys the bounds

[math]\displaystyle |\alpha(n)| \ll \tau^{O(1)}(n) \log^{O(1)}(x)[/math]
  • If [math]\alpha[/math] is a coefficient sequence and [math]a\ (q) = a \hbox{ mod } q[/math] is a primitive residue class, the (signed) discrepancy [math]\Delta(\alpha; a\ (q))[/math] of [math]\alpha[/math] in the sequence is defined to be the quantity
[math]\displaystyle \Delta(\alpha; a \ (q)) := \sum_{n: n = a\ (q)} \alpha(n) - \frac{1}{\phi(q)} \sum_{n: (n,q)=1} \alpha(n).[/math]
  • A coefficient sequence [math]\alpha[/math] is said to be at scale [math]N[/math] for some [math]N \geq 1[/math] if it is supported on an interval of the form [math][(1-O(\log^{-A_0} x)) N, (1+O(\log^{-A_0} x)) N][/math].
  • A coefficient sequence [math]\alpha[/math]at scale [math]N[/math] is said to obey the Siegel-Walfisz theorem if one has
[math] \displaystyle | \Delta(\alpha 1_{(\cdot,q)=1}; a\ (r)) | \ll \tau(qr)^{O(1)} N \log^{-A} x[/math]

for any [math]q,r \geq 1[/math], any fixed [math]A[/math], and any primitive residue class [math]a\ (r)[/math].

  • A coefficient sequence [math]\alpha[/math] at scale [math]N[/math] is said to be smooth if it takes the form [math]\alpha(n) = \psi(n/N)[/math] for some smooth function [math]\psi: {\mathbf R} \rightarrow {\mathbf C}[/math] supported on [math][1-O(\log^{-A_0} x), 1+O(\log^{-A_0} x)][/math] obeying the derivative bounds
[math]\displaystyle \psi^{(j)}(t) = O( \log^{j A_0} x ) [/math]

for all fixed [math]j \geq 0[/math] (note that the implied constant in the [math]O()[/math] notation may depend on [math]j[/math]).

Congruence class systems

Let [math]I \subset {\mathbf R}[/math], and let </math>{\mathcal S}_I</math> denote the square-free numbers whose prime factors lie in [math]I[/math].

  • A singleton congruence class system on [math]I[/math] is a collection [math]{\mathcal C} = (\{a_q\})_{q \in {\mathcal S}_I}[/math] of primitive residue classes [math]a_q \in ({\mathbf Z}/q{\mathbf Z})^\times[/math] for each </math>q \in {\mathcal S}_I</math>, obeying the Chinese remainder theorem property
[math]\displaystyle a_{qr}\ (qr) = (a_q\ (q)) \cap (a_r\ (r))[/math]

whenever [math]q,r \in {\mathcal S}_I[/math] are coprime. We say that such a system [math]{\mathcal C}[/math] has controlled multiplicity if the quantity

[math]\displaystyle \tau_{\mathcal C}(n) := |\{ q \in {\mathcal S}_I: n = a_q\ (q) \}|[/math]

obeys the estimate

[math]\displaystyle \sum_{C^{-1} x \leq n \leq Cx: n = a\ (r)} \tau_{\mathcal C}(n)^2 \ll \frac{x}{r} \tau(r)^{O(1)} \log^{O(1)} x + x^{o(1)}. [/math]

for any fixed [math]C \gt 1[/math] and any congruence class [math]a\ (r)[/math] with [math]r \in {\mathcal S}_I[/math]. Here [math]\tau[/math] is the divisor function.

Smooth and densely divisible numbers

A natural number [math]n[/math] is said to be [math]y[/math]-smooth if all of its prime factors are less than or equal to [math]y[/math]. We say that [math]n[/math] is [math]y[/math]-densely divisible</math> if, for every [math]1 \leq R \leq n[/math], one can find a factor of [math]n[/math] in the interval [math][y^{-1} R, R][/math].


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 ==