# Difference between revisions of "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 $Type_I[\varpi,\delta,\sigma]$, $Type_{II}[\varpi,\delta,\sigma]$, 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 $0 \lt \theta \lt 1/2$ be fixed.

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