\section{A giant prime factor in a fixed progression}
\label{sec:progressions}

The distribution of a large prime factor of $p-1$ will be needed with
$p$ restricted to a fixed residue class.  We prove this extension here.
The modulus remains fixed throughout the argument.

\begin{proposition}\label{prop:progressions}
Let $M\geq1$, let $U\subset(\Z/M\Z)^\times$, and put
\[
 B_U(n)=\mathbf1_{n\bmod M\in U},\qquad d_U=\frac{|U|}{\varphi(M)}.
\]
For the giant-factor statistic $f_x$ and reciprocal sum $C_x$ defined
above, and any fixed smooth function $\Phi$ of compact support in
$(0,\infty)$, one has
\begin{equation}\label{eq:ap-centered}
 \sum_{p\ {\rm prime}}\Phi\left(\frac{p-1}{x}\right)B_U(p)
       \bigl(f_x(p-1)-C_x\bigr)=o\left(\frac{x}{\log x}\right).
\end{equation}
Consequently,
\begin{equation}\label{eq:ap-giant}
 \sum_{p\ {\rm prime}}\Phi\left(\frac{p-1}{x}\right)B_U(p)f_x(p-1)
 =\left(d_Uc_\eps\int_0^\infty\Phi(t)\,dt+o(1)\right)
       \frac{x}{\log x}.
\end{equation}
All constants may depend on $M,U,\eps$, and $\Phi$.
\end{proposition}

The proof extends the prime extraction in \cite[Section~7]{PD}.
There are three points to check: the one-sided replacement of a prime
factor, the two-sided correlation after the small prime factors have
been removed, and the positive sieve used to remove the auxiliary
weights.  We give these checks explicitly.

\subsection{Character twists in the replacement estimates}
Retain the roughness threshold $w_x$ and the sieve product $V$ from
Section~\ref{sec:preliminaries}.
For a fixed Dirichlet character $\chi$ modulo $M$, the replacement
coefficient is
\begin{equation}\label{eq:ap-replacement-coefficient}
 \alpha_\chi(m)=\chi(m)m^{iv}
 \left(\mathbf1_{m\ {\rm prime}}-
       \frac{\mathbf1_{P^-(m)>w_x}}{V(w_x)\log m}\right),
\end{equation}
restricted to the interval occurring in the replacement.  Dirichlet
characters are extended by zero on nonunits.

The marked coefficient estimate \cite[Lemma~10.2]{AR} applies to
\eqref{eq:ap-replacement-coefficient}.  Indeed, testing this coefficient
against a character $\theta$ of modulus at most $L^A$ gives the original
prime-minus-rough coefficient tested against $\chi\theta$.  The latter
has a modulus at most $M L^A$, and hence at most $L^{A+1}$ for sufficiently
large $x$.  The long-coefficient estimate \cite[Lemma~5.2]{PD}, with this
larger fixed exponent, supplies every required logarithmic saving,
uniformly in the interval and the prescribed frequency range.  This
also applies to the conjugate coefficient.  Rough support and size
bounds are unchanged.  In particular, the second coefficient may be
multiplied by $\chi(n)$ as well.  The number of marking bands can be
chosen independently of their particular admissible exponents.

The corresponding two-sided assertion requires a separate argument:
a character of an integer need not be unchanged when an active prime
factor is removed.  We recall the notation needed for that argument.
Let $\mathcal P_g$ be the disjoint active prime groups in
\cite[Section~6]{PD}, let
\[
 V_g=\sum_{q\in\mathcal P_g}\frac1q,
 \qquad \omega_g(n)=\sum_{q\in\mathcal P_g}\mathbf1_{q\mid n},
 \qquad \omega_{\mathcal P}(n)=\sum_g\omega_g(n),
\]
and let $n_*$ be obtained from $n$ by removing all powers of primes in
$\mathcal P=\bigcup_g\mathcal P_g$.  There are $O(\log L)$ groups, each
with reciprocal mass bounded above and below by fixed positive
constants; the small groups lie between $\exp(L^{0.27})$ and
$\exp(L^{0.36})$, and the large groups between $\exp(L^{0.39})$ and
$\exp(L^{0.46})$.  For their fixed damping parameter $0<q_0<1$, put
\[
 A_{\mathbf t}(n)=q_0^{\omega_{\mathcal P}(n)-\sum_g t_g}
       \prod_g\frac{(\omega_g(n))_{t_g}}{V_g^{t_g}},
 \qquad A_1=A_{(1,\ldots,1)},
\]
where $(u)_t=u(u-1)\cdots(u-t+1)$, and the weight is zero if some
$\omega_g(n)<t_g$.  Bounded tests on the represented prime lists are
allowed in these weights.  For bounded $t_g$ they have size $L^{O(1)}$.
The fixed marking array $W$ uses smaller primes and is disjoint from
these groups.  Define
\[
 F_x(w)=W(w)\bigl(f_x(w)-C_x\bigr).
\]
Both $W$ and $f_x$ are invariant under multiplication by any active
prime, including its higher powers.  The same is therefore true of
$F_x$.

\begin{lemma}\label{lem:ap-two-sided}
Let $G(n)=G(n_*)$ and suppose
$|G(n)|\leq L^C\tau(n_*)^C$ for a fixed $C$.
For every fixed smooth compactly supported $\Psi$ and every $A>0$,
\begin{equation}\label{eq:ap-two-sided}
 \sum_{w\geq1}\frac{\Psi(w/x)}w F_x(w)\chi(w+1)G(w+1)
                 A_1(w)A_1(w+1)\ll L^{-A}.
\end{equation}
\end{lemma}

\begin{proof}
We modify the graph proof of \cite[Theorem~6.1]{PD}.  In its physical
coordinates an edge joins $n$ to $n'=n+D$, where $D=D_RD_C$ is the
product of its prime labels.  The factor $D_R$ consists of labels shared
by the two endpoints; $D_C$ consists of the remaining labels.  For large
$x$, every label is prime to $M$.

Use $\chi(n')G(n')$ at the second endpoint and multiply the edge by
$\overline{\chi(D)}$.  This modification is allowed by the graph
transference estimate \cite[Proposition~6.2]{PD}.  To see this, write
$e(t)=\exp(2\pi it)$ and expand
\[
 \overline{\chi(D)}=\sum_{r\bmod M}c_r e(rD/M),\qquad
 c_r=\frac1M\sum_{a\bmod M}\overline{\chi(a)}e(-ra/M).
\]
Parseval and Cauchy's inequality give $\sum_r|c_r|\leq\sqrt M$.
Since
\[
 e(rD/M)=e(rn'/M)e(-rn/M),
\]
each summand is the original graph pairing with both endpoint vectors
multiplied by $e(rn/M)$, using conjugation at the first endpoint in the
inner product.  These phases preserve all norms, supports, and
independence from the sampled prime labels.  The factor $\chi(n')$
can only decrease absolute values.  Thus the same transference bound
and the same finite family of comparison patterns
\cite[Proposition~6.3]{PD} apply.  Separating the smooth factor by Mellin
inversion commutes with this finite Fourier expansion.

Recombine the Fourier sum before removing the shared labels.  On
writing $n=D_Rw$ and $n'=D_R(w+D_C)$, the character factors become
\begin{equation}\label{eq:ap-shared-labels}
 \overline{\chi(D_RD_C)}\chi(D_R(w+D_C))
       =\overline{\chi(D_C)}\chi(w+D_C).
\end{equation}
This identity includes nonunits $w+D_C$, when both sides vanish.
The invariant cores can now be stripped exactly as in the original
proof.  Their reciprocal prime factors and damping weights are
unchanged; all collision errors have the original positive bounds.
The pattern with $D_C=1$ is precisely \eqref{eq:ap-two-sided}.

For a comparison pattern, $D_C=D_AD_B$, with the two free prime
products sampled independently.  Its separated expression has the
form
\[
 \mathbb E\, b_1(D_A)b_2(D_B)
       \sum_w\mathcal U(w)\mathcal V(w+D_AD_B),
 \qquad |b_1|,|b_2|\leq1.
\]
Here $\mathcal U$ and $\mathcal V$ are the first and second endpoint
sequences, including their smooth cutoffs and marked weights; the
expectation uses the reciprocal-prime measures of the two free lists.
Equation \eqref{eq:ap-shared-labels} replaces these three factors by
\[
 b_1(D_A)\overline{\chi(D_A)},\qquad
 b_2(D_B)\overline{\chi(D_B)},\qquad
 \chi(y)\mathcal V(y),
\]
respectively, and leaves $\mathcal U$ unchanged.  The comparison
estimates in \cite[Section~6]{PD} allow arbitrary bounded free-list
tests.  They use the second endpoint only through its global square
norm and coefficient mean-square bounds, which are preserved here.
Their local Fourier-energy and divisor-model cancellation estimates
act on the unchanged first endpoint.  Hence each comparison retains
its arbitrary logarithmic saving.  Subtracting the comparison terms
from the graph estimate proves the lemma.
\end{proof}

\subsection{The positive sieve in residue classes}
We may assume $\Phi\geq0$, by decomposing a general smooth function into
finitely many such functions with scalar coefficients.  Let $H$ be a
fixed marking-array weight, a product of two such weights, or a fixed
linear combination of these, and write $\mathbb E_x(H)$ for its
independent-divisor-model mean.  Choose $0<c_0<\eps/4$.
The progression estimate \cite[Lemma~7.2]{PD} has the following version:
\begin{align}
 \sum_{r\leq x^{c_0}}\Bigg|&
 \sum_{\substack{u\geq1\\r\mid u+1}}
       \Phi(u/x)B_U(u+1)f_x(u)H(u) \notag\\[-2mm]
 &-\frac{|U|}{M}\frac{\mathbf1_{(r,M)=1}}r\,
       xC_x\mathbb E_x(H)\int_0^\infty\Phi(t)\,dt
                   \Bigg|\ll_A xL^{-A}.
 \label{eq:ap-progression-estimate}
\end{align}
The same formula holds with $f_x$ and $C_x$ replaced by $1$.

Here is the modification of its proof.  Expand $H$ by the
represented-divisor expansion used in that lemma, keeping its positive
Bonferroni remainder.  Each retained divisor $d'$ is composed of
marking primes and has size $x^{o(1)}$; its total coefficient mass is
$x^{o(1)}$.  For a giant prime $\ell$ write $u=\ell y$.
Both $\ell$ and $d'$ are prime to the fixed $M$ for large $x$.
If $(r,M)>1$, no term is possible.  If $(r,M)=1$, the congruences for
a given $a\in U$ are
\[
 \ell y\equiv-1\pmod r,\qquad
 \ell y\equiv a-1\pmod M,\qquad d'\mid y.
\]
When $(d',r)=1$, smooth counting in the resulting progression gives
\[
 \frac{x}{Mr\ell d'}\int_0^\infty\Phi(t)\,dt+O_\Phi(1);
\]
otherwise there is no solution.  There is no need for $a-1$ to be a
unit modulo $M$.  The total rounding error is
$x^{1-\eps/2+c_0+o(1)}$, and the positive remainder and the exclusion of
marking primes dividing $r$ have the bounds of
\cite[Lemma~7.2]{PD}.  This proves
\eqref{eq:ap-progression-estimate} with the same model mean.

The local density of the condition $q\mid u+1$ in this calculation is
\[
 g(q)=\begin{cases}0,&q\mid M,\\1/q,&q\nmid M.\end{cases}
\]
For $z$ larger than the prime divisors of $M$,
\begin{equation}\label{eq:ap-sieve-density}
 \prod_{q\leq z}(1-g(q))=\frac{M}{\varphi(M)}V(z).
\end{equation}
The initial density $|U|/M$ therefore becomes $d_U$ at prime scale.
Apply the positive block sieve \cite[Lemma~2.6]{PD}, equivalently the
block construction of \cite[Lemma~2.9]{S}, with fixed even $h$ and
\[
 \gamma=\frac{c_0}{4h+3},\qquad z=x^\gamma.
\]
Its divisor level is at most $z^{4h+2}<x^{c_0}$, as required by
\eqref{eq:ap-progression-estimate}.  Subtracting $C_x$ times the
formula without $f_x$ cancels the main terms.  For a fixed nonnegative
array $W$, of mean $m_W$, this gives
\begin{equation}\label{eq:ap-presieve}
 \limsup_{x\to\infty}\frac{L}{xm_W}
 \left|\sum_{\substack{n\geq1\\P^-(n)>x^\gamma}}
 \Phi\left(\frac{n-1}{x}\right)B_U(n)W(n-1)
       \bigl(f_x(n-1)-C_x\bigr)\right|
       \ll \frac{e^{-h}}\gamma.
\end{equation}
The surviving constant may depend on the fixed progression and cutoff,
but not on the number or positions of the fixed marking bands.  The
factor $m_W$ occurs in both sieve main terms.  Errors depending on the
particular array disappear in the $x$-limit.

\subsection{Prime extraction and removal of the arrays}
Expand the residue-class indicator as
\begin{equation}\label{eq:ap-character-expansion}
 B_U(v)=\sum_{\chi\bmod M}b_\chi\chi(v),\qquad
 b_\chi=\frac1{\varphi(M)}\sum_{a\in U}\overline{\chi(a)}.
\end{equation}
The identity holds on nonunits as well.  In the composite removal of
\cite[Section~7]{PD}, a rough $v\asymp x$ is written as a product of a
bounded number of prime factors, the bound depending on $\gamma$.
Telescope each character term using
\eqref{eq:ap-replacement-coefficient}.  The other factors carry the
corresponding multiplicative character values.  All one-sided
replacement errors have arbitrary logarithmic saving, while the
predecessor core remains $f_x(v-1)-C_x$, clipped outside the required
range if necessary.  This clipping preserves its invariance under
marking primes.

In the subsequent allocation of active prime factors, retain
$\chi(v)$ as a factor of the whole integer $v$.  It is the same for
every allocation of a fixed factorization.  Thus the exact allocation
counts and damping identities are unchanged.  The errors in replacing
thresholds and logarithms by their stripped versions are bounded by the
same positive expressions as before, since $|\chi(v)|\leq1$; the
positive estimates controlling unsuccessful allocations are likewise
unchanged.  The remaining invariant core $G(v_*)$ is therefore the
original one.  Every final two-sided correlation is
\eqref{eq:ap-two-sided}.  Lemma~\ref{lem:ap-two-sided} bounds these
correlations with enough logarithmic saving to absorb the polynomial
in $L$ cost of the finite expansions.  Each composite contribution is
$o(x/L)$.  Consequently \eqref{eq:ap-presieve} also holds with its
rough sum replaced by the sum over primes.

For completeness, the last removal of the fixed arrays uses a positive
estimate.  Choose $b$ disjoint arrays $W_1,\ldots,W_b$, each with the
sufficient fixed number $K$ of bands, and let $m_j$ be their model means.
Put
\[
 Z_b(u)=\left|\frac1b\sum_{j=1}^b\frac{W_j(u)}{m_j}-1\right|^2.
\]
Apply \eqref{eq:ap-progression-estimate} without the giant factor to the
expansion of this nonnegative square, followed by the upper sieve.
The independent model for disjoint arrays, as in the deweighting step
of \cite[Section~7]{PD}, gives
\[
 \limsup_{x\to\infty}\frac{L}{x}
 \sum_{p\ {\rm prime}}\Phi\left(\frac{p-1}{x}\right)B_U(p)Z_b(p-1)
       \ll\frac{C_K}{b}.
\]
Cauchy's inequality, the boundedness of $f_x-C_x$ on the support, and
the prime-count upper bound now give
\[
 \limsup_{x\to\infty}\frac{L}{x}
 \left|\sum_{p\ {\rm prime}}\Phi\left(\frac{p-1}{x}\right)B_U(p)
                    \bigl(f_x(p-1)-C_x\bigr)\right|
       \ll \frac{e^{-h}}\gamma+\sqrt{\frac{C_K}{b}}.
\]
First fix $h,\gamma,K,b$ and the arrays, and take $x\to\infty$.
Next let $b\to\infty$ with $h,K$ fixed.  Finally let the even integer
$h\to\infty$, choosing a sufficient $K$ each time.  Since
$\gamma^{-1}=(4h+3)/c_0$, both errors vanish.  This proves
\eqref{eq:ap-centered}.  The prime number theorem in the fixed residue
classes, together with $C_x\to c_\eps$, proves
\eqref{eq:ap-giant} and completes the proof of
Proposition~\ref{prop:progressions}.