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

## Definitions

### Asymptotic notation

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

### Coefficient sequences

We need a fixed quantity $A_0\gt0$.

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

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

for any $q,r \geq 1$, any fixed $A$, and any primitive residue class $a\ (r)$.

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

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

### Congruence class systems

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

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

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

$\displaystyle \tau_{\mathcal C}(n) := |\{ q \in {\mathcal S}_I: n = a_q\ (q) \}|$

obeys the estimate

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

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

### Smooth and densely divisible numbers

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

### MPZ

Let $0 \lt \varpi \lt 1/4$ and $0 \lt \delta \lt \varpi + 1/4$ be fixed. Let $\Lambda$ denote the von Mangoldt function.

• We say that the estimate $MPZ[\varpi,\delta]$ holds if one has the estimate
$\displaystyle \sum_{q \in {\mathcal S}_I: q\lt x^{1/2+2\varpi}} |\Delta(\Lambda 1_{[x,2x]}; a_q)| \ll x \log^{-A} x$

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

• We say that the estimate $MPZ'[\varpi,\delta]$ holds if one has the estimate
$\displaystyle \sum_{q \in {\mathcal S}_I \cap {\mathcal D}_{x^\delta}: q\lt x^{1/2+2\varpi}} |\Delta(\Lambda 1_{[x,2x]}; a_q)| \ll x \log^{-A} x$

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

### Type I, Type II, and Type III

Let $0 \lt \varpi \lt 1/4$, $0 \lt \delta \lt 1/4+\varpi$, and $0 \lt \sigma \lt 1/2$ be fixed.

• We say that $Type_I[\varpi,\delta,\sigma]$ holds if, whenever $M,N$ are quantities with
$\displaystyle MN \sim x$

and

$\displaystyle x^{1/2-\sigma} \ll N \ll x^{1/2-2\varpi-c}$

or equivalently

$\displaystyle x^{1/2+2\varpi+c} \ll M \ll x^{1/2+\sigma}$

for some fixed $c\gt0$, and $\alpha,\beta$ are coefficient sequences at scale $M,N$ respectively with $\beta$ obeying a Siegel-Walfisz theorem, $I \subset [1,x^\delta]$, and $(\{a_q\})_{q \in {\mathcal S}_I}$ is a congruence class system of controlled multiplicity, then one has

$\sum_{q \in {\mathcal S}_I: q \lt x^{1/2+2\varpi}} |\Delta( \alpha * \beta; a_q\ (q))| \leq x \log^{-A} x$

for all fixed $A\gt0$.

• We say that $Type_{II}[\varpi,\delta]$ holds if, whenever $M,N$ are quantities with
$\displaystyle MN \sim x$

and

$\displaystyle x^{1/2-2\varpi-c} \ll N \ll x^{1/2}$

or equivalently

$\displaystyle x^{1/2} \ll M \ll x^{1/2+2\varpi+c}$

for some sufficiently small fixed $c\gt0$, and $\alpha,\beta$ are coefficient sequences at scale $M,N$ respectively with $\beta$ obeying a Siegel-Walfisz theorem, $I \subset [1,x^\delta]$, and $(\{a_q\})_{q \in {\mathcal S}_I}$ is a congruence class system of controlled multiplicity, then one has

$\sum_{q \in {\mathcal S}_I: q \lt x^{1/2+2\varpi}} |\Delta( \alpha * \beta; a_q\ (q))| \leq x \log^{-A} x$

for all fixed $A\gt0$.

• We say that $Type_{III}[\varpi,\delta,\sigma]$ holds if, whenever $M,N_1,N_2,N_3$ are quantities with
$\displaystyle MN \sim x$
$\displaystyle N_1N_2, N_2 N_3, N_1 N_3 \gg x^{1/2 + \sigma}$
$\displaystyle x^{2\sigma} \ll N_1,N_2,N_3 \ll x^{1/2-\sigma},$

$\alpha,\psi_1,\psi_2,\psi_3$ are coefficient sequences at scale $M,N_1,N_2,N_3$ respectively with $\psi_1,\psi_2,\psi_3$ smooth, $I \subset [1,x^\delta]$, and $(\{a_q\})_{q \in {\mathcal S}_I}$ is a congruence class system of controlled multiplicity, then one has

$\sum_{q \in {\mathcal S}_I: q \lt x^{1/2+2\varpi}} |\Delta( \alpha * \psi_1 * \psi_2 * \psi_3; a_q\ (q))| \leq x \log^{-A} x$

for all fixed $A\gt0$.

• We define $Type'_I[\varpi,\delta,\sigma]$, $Type'_{II}[\varpi,\delta]$, $Type_{III}[\varpi,\delta,\sigma]$ analogously to $Type_I[\varpi,\delta,\sigma]$, $Type_{II}[\varpi,\delta]$, $Type_{III}[\varpi,\delta,\sigma]$ but with the hypothesis $I \subset [1,x^\delta]$ replaced with $I \subset \mathbf{R}$, and ${\mathcal S}_I$ replaced with ${\mathcal S}_I \cap {\mathcal D}_{x^\delta}$. These estimates are slightly stronger than their unprimed counterparts.

There should also be a second "double-primed" variant $Type''_I[\varpi,\delta,\sigma], Type''_{II}[\varpi,\delta], Type''_{III}[\varpi,\delta,\sigma]$ of these estimates, intermediate in strength between the primed and unprimed estimates, in which one assumes a suitable "double dense divisibility" hypothesis, which has not yet been determined precisely.

Note: thus far in the Type III analysis, the controlled multiplicity hypothesis has yet to be used.

## The combinatorial lemma

Combinatorial lemma Let $0 \lt \varpi \lt 1/4$, $0 \lt \delta \lt 1/4 + \varpi$, and $1/10 \lt \sigma \lt 1/12$ be fixed.
• If $Type_I[\varpi,\delta,\sigma]$, $Type_{II}[\varpi,\delta]$, and $Type_{III}[\varpi,\delta,\sigma]$ all hold, then $MPZ[\varpi,\delta]$ holds.
• Similarly, if $Type'_I[\varpi,\delta,\sigma]$, $Type'_{II}[\varpi,\delta]$, and $Type'_{III}[\varpi,\delta,\sigma]$ all hold, then $MPZ'[\varpi,\delta]$ holds.

This lemma is (somewhat implicitly) proven here. It reduces the verification of $MPZ[\varpi,\delta]$ and $MPZ'[\varpi,\delta]$ to a comparison of the best available Type I, Type II, and Type III estimates, as well as the constraint $\sigma \gt 1/10$.

## Type I estimates

In all of the estimates below, $0 \lt \varpi \lt 1/4$, $0 \lt \delta \lt 1/4 + \varpi$, and $\sigma \gt 0$ are fixed.

### Level 1

Type I-1 We have $Type'_I[\varpi,\delta,\sigma]$ (and hence $Type_I[\varpi,\delta,\sigma]$) whenever
$\displaystyle 11\varpi +3\delta + 2 \sigma \lt \frac{1}{4}$.

This result is implicitly proven here. (There, only $Type_I[\varpi,\delta,\sigma]$ is proven, but the method extends without difficulty to $Type'_I[\varpi,\delta,\sigma]$.) It uses the method of Zhang, and is ultimately based on exponential sums for incomplete Kloosterman sums on smooth moduli obtained via completion of sums.

### Level 2

Type I-2 We have $Type'_I[\varpi,\delta,\sigma]$ (and hence $Type_I[\varpi,\delta,\sigma]$) whenever
$\displaystyle 14\varpi +4\delta + \sigma \lt \frac{1}{4}$

and

$\displaystyle 20\varpi +6\delta + 3\sigma \lt \frac{1}{2}$

and

$\displaystyle 32\varpi +9\delta + \sigma \lt \frac{1}{2}$.

This estimate is implicitly proven here. It improves upon the Level 1 estimate by using the q-van der Corput A-process in the $d_2$ direction.

### Level 3

Type I-3 We have $Type'_I[\varpi,\delta,\sigma]$ (and hence $Type_I[\varpi,\delta,\sigma]$) whenever
$\displaystyle 54\varpi + 15 \delta + 5 \sigma \lt 1$

and

$\displaystyle 32\varpi +9\delta + \sigma \lt \frac{1}{2}$.

This estimate is tentatively established in this comment. It improves upon the Level 2 estimate by taking advantage of dense divisibility to optimise the direction of averaging.

### Level 4

By iterating the q-van der Corput A-process, one should be able to obtain $Type''_I[\varpi,\delta,\sigma]$ assuming a constraint of the form

$\displaystyle 40\varpi + C \delta + 4 \sigma \lt 1$

for some constant C that has not yet been determined (in part because we have not yet decided what "doubly densely divisible" means); see this comment.

### Level 5

Further improvement to the (still sketchy) Level 4 estimate should be obtainable by taking advantage of averaging in auxiliary "h" parameters in order to reduce the contribution of the diagonal terms.

## Type II estimates

In all of the estimates below, $0 \lt \varpi \lt 1/4$ and $0 \lt \delta \lt 1/4 + \varpi$ are fixed.

### Level 1

Type II-1 We have $Type'_{II}[\varpi,\delta]$ (and hence $Type_{II}[\varpi,\delta]$) whenever
$\displaystyle 58\varpi + 10\delta \lt \frac{1}{2}$.

This estimate is implicitly proven here. (There, only $Type_I[\varpi,\delta,\sigma]$ is proven, but the method extends without difficulty to $Type'_I[\varpi,\delta,\sigma]$.) It uses the method of Zhang, and is ultimately based on exponential sums for incomplete Kloosterman sums on smooth moduli obtained via completion of sums.

### Level 1a

Type II-1a We have $Type'_{II}[\varpi,\delta]$ (and hence $Type_{II}[\varpi,\delta]$) whenever
$\displaystyle 48\varpi + 7\delta \lt \frac{1}{2}$.

This estimate is implicitly proven here. It is a slight refinement of the Level 1 estimate based on a more careful inspection of the error terms in the completion of sums method.

### Level 2

In analogy with the Type I-2 estimates, one should be able to improve the Type II estimates by using the q-van der Corput process in the $d_2$ direction.

### Level 3

In analogy with the Type I-3 estimates, one should be able to improve the Type II estimates by using the q-van der Corput process in an optimised direction.

### Level 4

In analogy with the Type I-4 estimates, one should be able to improve the Type II estimates by iterating the q-van der Corput A-process.

### Level 5

In analogy with the Type I-5 estimates, one should be able to improve the Type II estimates by taking advantage of averaging in the h parameters.

## Type III estimates

In all of the estimates below, $0 \lt \varpi \lt 1/4$, $0 \lt \delta \lt 1/4 + \varpi$, and $\sigma \gt 0$ are fixed.

### Level 1

Type III-1 We have $Type'_{III}[\varpi,\delta,\sigma]$ (and hence $Type_{III}[\varpi,\delta,\sigma]$) whenever
$\displaystyle \frac{13}{2} (\frac{1}{2} + \sigma) \gt 8 (\frac{1}{2} + 2 \varpi) + \delta$

This estimate is implicitly proven here. (There, only $Type_{III}[\varpi,\delta,\sigma]$ is proven, but the method extends without difficulty to $Type'_{III}[\varpi,\delta,\sigma]$.) It uses the method of Zhang, using Weyl differencing and not exploiting the averaging in the $\alpha$ or $q$ parameters. The constraint can also be written as a lower bound on $\sigma$:

$\displaystyle \sigma \gt \frac{3}{26} + \frac{32}{13} \varpi + \frac{2}{13} \delta$.

### Level 2

Type III-2 We have $Type'_{III}[\varpi,\delta,\sigma]$ (and hence $Type_{III}[\varpi,\delta,\sigma]$) whenever
$\displaystyle 1 + 5 (\frac{1}{2} + \sigma) \gt 8 (\frac{1}{2} + 2 \varpi) + \delta$

This estimate is implicitly proven here. It is a refinement of the Level 1 estimate that takes advantage of the $\alpha$ averaging. The constraint may also be written as a lower bound on $\sigma$:

$\displaystyle \sigma \gt \frac{1}{10} + \frac{16}{5} \varpi + \frac{1}{5} \delta$.

### Level 3

Type III-3 We have $Type'_{III}[\varpi,\delta,\sigma]$ (and hence $Type_{III}[\varpi,\delta,\sigma]$) whenever
$\displaystyle 3 (\frac{1}{2} + \sigma) \gt \frac{7}{4} (\frac{1}{2} + 2 \varpi) + \frac{3}{8} \delta$.

This estimate is proven in this comment. It uses the newer method of Fouvry, Kowalski, and Michel that avoids Weyl differencing. The constraint may also be written as a lower bound on $\sigma$:

$\displaystyle \sigma \gt \frac{1}{12} + \frac{7}{3} \varpi + \frac{1}{4} \delta$.

### Level 4

It should be possible to improve upon the Level 3 estimate by exploiting averaging in the $\alpha$ parameter.