\section{Divisor weights and the marked bilinear estimate}
\label{sec:preliminaries}

We use finite collections of small prime divisors to insert weights into
bilinear sums.  We give the definitions and the precise external estimate
needed below.  All numbers of bands and all band exponents in this section
are fixed before $x$ tends to infinity.

Put
\[
 L=\log x,\qquad w_x=\exp(L^{0.24}),\qquad
 V(y)=\prod_{p\leq y}\left(1-\frac1p\right).
\]
We write $P^-(n)$ for the least prime divisor of $n>1$, and put
$P^-(1)=\infty$.  A positive integer $n$ is $y$-rough if $P^-(n)>y$.
Dirichlet characters, including principal characters, are extended by
zero on integers not coprime to their modulus.

\subsection{The divisor weights}

Fix $K\geq1$ and
$0.1<\alpha_1<\cdots<\alpha_K<0.2$.  Set
\[
 \mathcal P_i=\{p\text{ prime}:\exp(L^{\alpha_i})\leq p\leq
                 \exp(2L^{\alpha_i})\},\qquad
 V_i=\sum_{p\in\mathcal P_i}\frac1p,
 \qquad
 \omega_i(u)=\sum_{p\in\mathcal P_i}\mathbf1_{p\mid u}.
\]
The bands $\mathcal P_i$ are disjoint for large $x$, and
$V_i=\log2+o(1)$.  With $\rho=1/2$, define
\begin{equation}\label{eq:array-weight}
 W(u)=\prod_{i=1}^K
       \frac{\omega_i(u)\rho^{\omega_i(u)-1}}{V_i}.
\end{equation}
The factor is zero when $\omega_i(u)=0$.  We call the collection of
bands, together with this weight, an \emph{array}.  The weight is
nonnegative, supported on integers having a prime divisor in every
band, and bounded by a constant depending only on $K$ for large $x$.
Equivalently, with $\omega_{\rm band}(u)=\sum_i\omega_i(u)$,
\[
 W(u)=\rho^{\omega_{\rm band}(u)-K}
 \sum_{\substack{p_i\in\mathcal P_i\ (1\leq i\leq K)\\
                  p_1\cdots p_K\mid u}}
       \prod_{i=1}^K V_i^{-1}.
\]
This identity specifies the normalization when one prime divisor is
selected from each band.  In particular, the sum over selected divisors
is a counting sum; the reciprocal factors $1/p_i$ enter its mean.

The \emph{independent divisor model} replaces the indicators
$\mathbf1_{p\mid u}$, for all primes in the arrays under consideration,
by independent Bernoulli variables of means $1/p$.  Denote its
expectation by $\mathbb E_x$, and put
\begin{equation}\label{eq:array-mean}
 m_W=\mathbb E_x W.
\end{equation}
For example, independence between bands gives the exact formula
\[
 m_W=\prod_{i=1}^K
 \left\{
 \prod_{p\in\mathcal P_i}\left(1-\frac{1-\rho}{p}\right)
 \frac1{V_i}\sum_{p\in\mathcal P_i}\frac1{p-(1-\rho)}
 \right\}.
\]
Consequently, for constants depending on $K$ but not on the fixed band
exponents,
\begin{equation}\label{eq:array-moments}
 0<c_K\leq m_W\leq C_K,
 \qquad \mathbb E_x(W^2)\leq C_K
\end{equation}
for all sufficiently large $x$.  The upper bounds also follow directly
from boundedness of $W$.  For the lower bound, the event of exactly one
divisor in each band has probability bounded below, since
$V_i\longrightarrow\log2$ and
$\sum_{p\in\mathcal P_i}p^{-2}\longrightarrow0$.

Let
\[
 u_{\mathcal P}=\prod_{p\in\bigcup_i\mathcal P_i}p^{v_p(u)}.
\]
A function $F:\N\to\C$ is \emph{invariant under the band primes} if
\begin{equation}\label{eq:band-invariance}
 F(pu)=F(u)\quad
 (u\geq1,\ p\in\bigcup_i\mathcal P_i).
\end{equation}
Equivalently, $F(u)$ depends only on $u/u_{\mathcal P}$.  This condition
removes the whole band-prime part, including repeated prime factors.
The centered large-prime statistic $f_x-C_x$ is invariant in this sense
for sufficiently large $x$.

\subsection{The coefficient criterion}

The following is the coefficient criterion of
\cite[Lemma~10.2]{AR}.  Its important uniformity is that the required
number $K$ of bands is independent of their particular fixed exponents.

\begin{proposition}[Marked bilinear estimate]\label{prop:transfer}
Fix $\delta,C,D>0$.  Suppose that $H_m,H_n\geq x^\delta$ and
$X=H_mH_n\asymp x$, with fixed comparison constants.  Let $F$ satisfy
$|F|\leq1$ and \eqref{eq:band-invariance}.  Let $\alpha_m,\beta_n$ be
supported on $w_x$-rough integers in $[H_m,2H_m]$ and $[H_n,2H_n]$,
respectively, with $|\alpha_m|,|\beta_n|\leq L^C$.  The support of
$\alpha$ may be restricted to any interval $I\subseteq[H_m,2H_m]$.
Assume that, for every fixed $A_0,B,A>0$, every Dirichlet character
$\vartheta$ modulo $k\leq L^{A_0}$, and every $|t|\leq2XL^B$,
\begin{equation}\label{eq:coefficient-test}
 \frac1{H_m}\left|\sum_m\alpha_m\vartheta(m)m^{it}\right|
       \ll_A L^{-A},
\end{equation}
uniformly in the interval, character, and frequency.  Constants in this
hypothesis may depend on all fixed parameters defining the coefficient.
Then, for sufficiently large fixed $K$,
\begin{equation}\label{eq:marked-bilinear}
 \left|\sum_{m,n}\alpha_m\beta_n F(mn-1)W(mn-1)\right|
       \ll X L^{-D}.
\end{equation}
The required lower bound on $K$ is independent of the band exponents
$\alpha_i$.  The implicit constant and the threshold for $x$ may depend on
those exponents and on all other fixed data.
\end{proposition}

The version with prime-minus-rough coefficients is
\cite[Theorem~3.1]{PD}.  In particular, the sequence
\begin{equation}\label{eq:prime-rough-coefficient}
 \alpha_m=m^{iv}
 \left(\mathbf1_{m\text{ prime}}
       -\frac{\mathbf1_{P^-(m)>w_x}}{V(w_x)\log m}\right),
 \qquad |v|\leq L^C,
\end{equation}
restricted to a permitted interval, satisfies
\eqref{eq:coefficient-test} by \cite[Lemma~5.2]{PD}.
Multiplying this coefficient by a fixed Dirichlet character preserves
the criterion: the product with a test character has modulus at most
$M L^{A_0}\leq L^{A_0+1}$ for all sufficiently large $x$, where $M$ is
fixed.  This observation will be used for fixed arithmetic progressions.

\subsection{Removing the divisor weights at primes}

We also record the positive estimate that will remove these weights.
Choose $b$ arrays $W_1,\ldots,W_b$, each with $K$ bands, so that all
their band exponents are distinct.  Write $m_j=\mathbb E_xW_j$ and
\[
 Z_b(u)=\left(\frac1b\sum_{j=1}^b\frac{W_j(u)}{m_j}-1\right)^2.
\]
Disjoint arrays are independent in the divisor model.  Thus
\begin{equation}\label{eq:model-array-variance}
 \mathbb E_x Z_b
 =\frac1{b^2}\sum_{j=1}^b
       \operatorname{Var}_x(W_j/m_j)
 \leq \frac{C_K}{b}.
\end{equation}
For every fixed nonnegative smooth compactly supported $\Phi$ on
$(0,\infty)$, the prime version is
\begin{equation}\label{eq:array-variance}
 \limsup_{x\to\infty}\frac{L}{x}
 \sum_{p\text{ prime}}\Phi((p-1)/x)Z_b(p-1)
       \ll_\Phi\frac{C_K}{b}.
\end{equation}
The constant is independent of $b$ and of the fixed band exponents.
Here and below $b$ is fixed while taking the limit in $x$.

For completeness, this estimate uses only the progression calculation
and the positive sieve in \cite[\S7]{PD}.  The summed progression
estimate of \cite[Lemma~7.2]{PD}, with the large-prime statistic omitted,
applies to $Z_b$: its expansion contains constants, single arrays, and
products of two arrays, including repeated arrays.  Apply the upper
bound in the block sieve of \cite[Lemma~2.6]{PD} at depth $2$, with
$z=x^{c_0/20}$ for a fixed permitted progression exponent $c_0>0$.
Its divisor level is $z^{10}=x^{c_0/2}$, inside that progression
range.  Every prime $p$ in the support exceeds $z$, so its contribution
is bounded by the sifted nonnegative weight.  The main term is
$O_\Phi(x\mathbb E_xZ_b/\log z)$, and the summed remainder is
$o(x/L)$.  Equation~\eqref{eq:model-array-variance} proves
\eqref{eq:array-variance}.

In particular, if $|a_p|\leq C$ at primes, Cauchy's inequality and the
prime number theorem give
\begin{equation}\label{eq:array-removal}
 \limsup_{x\to\infty}\frac Lx
 \left|\sum_p\Phi((p-1)/x)a_p
       \left(\frac1b\sum_{j=1}^b\frac{W_j(p-1)}{m_j}-1\right)\right|
       \ll_{\Phi,C}\sqrt{\frac{C_K}{b}}.
\end{equation}
This estimate concerns bounded prime coefficients; it requires no
bound for their extension to composite integers.