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].

Definitions

Asymptotic notation

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

Coefficient sequences

We need a fixed quantity [math]\displaystyle{ A_0\gt 0 }[/math].

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

[math]\displaystyle{ \displaystyle |\alpha(n)| \ll \tau^{O(1)}(n) \log^{O(1)}(x) }[/math]

• If [math]\displaystyle{ \alpha }[/math] is a coefficient sequence and [math]\displaystyle{ a\ (q) = a \hbox{ mod } q }[/math] is a primitive residue class, the (signed) discrepancy [math]\displaystyle{ \Delta(\alpha; a\ (q)) }[/math] of [math]\displaystyle{ \alpha }[/math] in the sequence is defined to be the quantity

[math]\displaystyle{ \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]\displaystyle{ \alpha }[/math] is said to be at scale [math]\displaystyle{ N }[/math] for some [math]\displaystyle{ N \geq 1 }[/math] if it is supported on an interval of the form [math]\displaystyle{ [(1-O(\log^{-A_0} x)) N, (1+O(\log^{-A_0} x)) N] }[/math].

• A coefficient sequence [math]\displaystyle{ \alpha }[/math] at scale [math]\displaystyle{ N }[/math] is said to obey the Siegel-Walfisz theorem if one has

[math]\displaystyle{ \displaystyle | \Delta(\alpha 1_{(\cdot,q)=1}; a\ (r)) | \ll \tau(qr)^{O(1)} N \log^{-A} x }[/math]

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

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

[math]\displaystyle{ \displaystyle \psi^{(j)}(t) = O( \log^{j A_0} x ) }[/math]

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

Congruence class systems

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

• A singleton congruence class system on [math]\displaystyle{ I }[/math] is a collection [math]\displaystyle{ {\mathcal C} = (\{a_q\})_{q \in {\mathcal S}_I} }[/math] of primitive residue classes [math]\displaystyle{ 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{ \displaystyle a_{qr}\ (qr) = (a_q\ (q)) \cap (a_r\ (r)) }[/math]

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

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

obeys the estimate

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

Smooth and densely divisible numbers

A natural number [math]\displaystyle{ n }[/math] is said to be [math]\displaystyle{ y }[/math]-smooth if all of its prime factors are less than or equal to [math]\displaystyle{ y }[/math]. We say that [math]\displaystyle{ n }[/math] is [math]\displaystyle{ y }[/math]-densely divisible if, for every [math]\displaystyle{ 1 \leq R \leq n }[/math], one can find a factor of [math]\displaystyle{ n }[/math] in the interval [math]\displaystyle{ [y^{-1} R, R] }[/math]. Note that [math]\displaystyle{ y }[/math]-smooth numbers are automatically [math]\displaystyle{ y }[/math]-densely divisible, but the converse is not true in general.

MPZ

Let [math]\displaystyle{ 0 \lt \varpi \lt 1/4 }[/math] and [math]\displaystyle{ 0 \lt \delta \lt \varpi + 1/4 }[/math] be fixed. Let [math]\displaystyle{ \Lambda }[/math] denote the von Mangoldt function.

• We say that the estimate [math]\displaystyle{ MPZ[\varpi,\delta] }[/math] holds if one has the estimate

[math]\displaystyle{ \displaystyle \sum_{q \in {\mathcal S}_I: q\lt x^{1/2+2\varpi}} |\Delta(\Lambda 1_{[x,2x]}; a_q)| \ll x \log^{-A} x }[/math]

for any fixed [math]\displaystyle{ A \gt 0 }[/math], any [math]\displaystyle{ I \subset [1,x^\delta] }[/math], and any congruence class system [math]\displaystyle{ (\{a_q\})_{q \in {\mathcal S}_I} }[/math] of controlled multiplicity.

• We say that the estimate [math]\displaystyle{ MPZ'[\varpi,\delta] }[/math] holds if one has the estimate

[math]\displaystyle{ \displaystyle \sum_{q \in {\mathcal S}_I \cap {\mathcal D}_{x^\delta}: q\lt x^{1/2+2\varpi}} |\Delta(\Lambda 1_{[x,2x]}; a_q)| \ll x \log^{-A} x }[/math]

for any fixed [math]\displaystyle{ A \gt 0 }[/math], any [math]\displaystyle{ I \subset {\mathbf R} }[/math], and any congruence class system [math]\displaystyle{ (\{a_q\})_{q \in {\mathcal S}_I} }[/math] of controlled multiplicity.

Type I, Type II, and Type III

Let [math]\displaystyle{ 0 \lt \varpi \lt 1/4 }[/math], [math]\displaystyle{ 0 \lt \delta \lt 1/4+\varpi }[/math], and [math]\displaystyle{ 0 \lt \theta \lt 1/2 }[/math] be fixed.

• We say that [math]\displaystyle{ Type_I[\varpi,\delta,\theta] }[/math] holds if

• We say that [math]\displaystyle{ Type_{II}[\varpi,\delta] }[/math] holds if

• We say that [math]\displaystyle{ Type_{III}[\varpi,\delta,\theta] }[/math] holds if

• We define [math]\displaystyle{ Type'_I[\varpi,\delta,\theta] }[/math], [math]\displaystyle{ Type'_{II}[\varpi,\delta] }[/math], [math]\displaystyle{ Type_{III}[\varpi,\delta,\theta] }[/math] analogously to [math]\displaystyle{ Type_I[\varpi,\delta,\theta] }[/math], [math]\displaystyle{ Type_{II}[\varpi,\delta] }[/math], [math]\displaystyle{ Type_{III}[\varpi,\delta,\theta] }[/math] but with the hypothesis [math]\displaystyle{ I \subset [1,x^\delta] }[/math] replaced with [math]\displaystyle{ I \subset \mathbf{R} }[/math], and [math]\displaystyle{ {\mathcal S}_I }[/math] replaced with [math]\displaystyle{ {\mathcal S}_I \cap {\mathcal D}_{x^\delta} }[/math].