==== TABLE OF CONTENTS ====
Simultaneous-primitive-roots-a-conditional-lower-bound-for-prime-bases//paper.tex
Simultaneous-primitive-roots-a-conditional-lower-bound-for-prime-bases//references.tex
Simultaneous-primitive-roots-a-conditional-lower-bound-for-prime-bases//sections/00-abstract.tex
Simultaneous-primitive-roots-a-conditional-lower-bound-for-prime-bases//sections/01-introduction.tex
Simultaneous-primitive-roots-a-conditional-lower-bound-for-prime-bases//sections/02-reduction.tex
Simultaneous-primitive-roots-a-conditional-lower-bound-for-prime-bases//sections/03-splitting.tex
Simultaneous-primitive-roots-a-conditional-lower-bound-for-prime-bases//sections/04-marked-family.tex
Simultaneous-primitive-roots-a-conditional-lower-bound-for-prime-bases//sections/05-prime-reservoir.tex
===========================
==== BEGIN Simultaneous-primitive-roots-a-conditional-lower-bound-for-prime-bases//paper.tex ====
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\hypersetup{
pdftitle={Simultaneous primitive roots: a conditional lower bound for prime bases},
pdfauthor={OpenAI},
pdfsubject={A conditional lower bound for primes with prescribed simultaneous primitive roots},
pdfkeywords={primitive roots, Artin conjecture, sieve, Kummer extensions, Hecke L-functions}
}
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\newtheorem{theorem}{Theorem}[section]
\newtheorem{proposition}[theorem]{Proposition}
\newtheorem{lemma}[theorem]{Lemma}
\newtheorem{corollary}[theorem]{Corollary}
\theoremstyle{definition}
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\newtheorem{definition}[theorem]{Definition}
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\newcommand{\one}{\mathbf 1}
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\title{Simultaneous primitive roots: a conditional lower bound for prime bases}
\author{OpenAI}
\date{October 4, 2026}
\begin{document}
\maketitle
\input{sections/00-abstract}
\input{sections/01-introduction}
\input{sections/02-reduction}
\input{sections/03-splitting}
\input{sections/04-marked-family}
\input{sections/05-prime-reservoir}
\input{references}
\end{document}
==== END Simultaneous-primitive-roots-a-conditional-lower-bound-for-prime-bases//paper.tex ====
==== BEGIN Simultaneous-primitive-roots-a-conditional-lower-bound-for-prime-bases//references.tex ====
\begin{thebibliography}{99}
\raggedright
\bibitem{AnwarPappalardi2017}
M.~Anwar and F.~Pappalardi,
On simultaneous primitive roots,
\emph{Acta Arith.} \textbf{180} (2017), no.~1, 35--43.
\href{https://doi.org/10.4064/aa8566-3-2017}{doi:10.4064/aa8566-3-2017}.
\bibitem{Bombieri1965}
E.~Bombieri,
On the large sieve,
\emph{Mathematika} \textbf{12} (1965), no.~2, 201--225.
\href{https://doi.org/10.1112/S0025579300005313}{doi:10.1112/S0025579300005313}.
\bibitem{GuptaMurty1984}
R.~Gupta and M.~Ram Murty,
A remark on Artin's conjecture,
\emph{Invent. Math.} \textbf{78} (1984), no.~1, 127--130.
\href{https://doi.org/10.1007/BF01388719}{doi:10.1007/BF01388719}.
\bibitem{HasseDiaries2012}
H.~Hasse,
\emph{Die mathematischen Tageb\"ucher von Helmut Hasse 1923--1935},
edited by F.~Lemmermeyer and P.~Roquette,
Universit\"atsverlag G\"ottingen, 2012.
\href{https://univerlag.uni-goettingen.de/handle/3/isbn-978-3-86395-072-9}{Publisher's edition}.
\bibitem{HasanalizadeShenWong2022}
E.~Hasanalizade, Q.~Shen, and P.-J.~Wong,
Counting zeros of Dedekind zeta functions,
\emph{Math. Comp.} \textbf{91} (2022), no.~333, 277--293.
\href{https://doi.org/10.1090/mcom/3665}{doi:10.1090/mcom/3665}.
\bibitem{HeathBrown1986}
D.~R.~Heath-Brown,
Artin's conjecture for primitive roots,
\emph{Quart. J. Math. Oxford} (2) \textbf{37} (1986), no.~1, 27--38.
\href{https://doi.org/10.1093/qmath/37.1.27}{doi:10.1093/qmath/37.1.27}.
\bibitem{Hooley1967}
C.~Hooley,
On Artin's conjecture,
\emph{J. Reine Angew. Math.} \textbf{225} (1967), 209--220.
\href{https://doi.org/10.1515/crll.1967.225.209}{doi:10.1515/crll.1967.225.209}.
\bibitem{Matthews1976}
K.~Matthews,
A generalisation of Artin's conjecture for primitive roots,
\emph{Acta Arith.} \textbf{29} (1976), no.~2, 113--146.
\href{https://doi.org/10.4064/aa-29-2-113-146}{doi:10.4064/aa-29-2-113-146}.
\bibitem{Neukirch1999}
J.~Neukirch,
\emph{Algebraic Number Theory},
translated by N.~Schappacher,
Grundlehren der mathematischen Wissenschaften, vol.~322,
Springer, Berlin, 1999.
\href{https://doi.org/10.1007/978-3-662-03983-0}{doi:10.1007/978-3-662-03983-0}.
\bibitem[P]{PrimitiveRoot2}
OpenAI,
Primitive roots for every admissible integer base,
OpenAI Math Release preprint
\href{https://github.com/openai/math/blob/main/preprints/Primitive-roots-for-every-admissible-integer-base-October-4-2026/primitive-roots-all-integer-bases.pdf}{OAI:Primitive-roots-for-every-admissible-integer-base-October-4-2026},
2026.
\bibitem{Tate1967}
J.~T.~Tate,
Fourier analysis in number fields and Hecke's zeta-functions,
in \emph{Algebraic Number Theory}, edited by J.~W.~S.~Cassels and
A.~Fr\"ohlich, Academic Press, London, 1967, 305--347.
\href{https://math.arizona.edu/~cais/scans/Cassels-Frohlich-Algebraic_Number_Theory.pdf}{Scan of the volume}.
\bibitem{Vinogradov1965}
A.~I.~Vinogradov,
On the density hypothesis for Dirichlet \(L\)-series,
\emph{Izv. Akad. Nauk SSSR Ser. Mat.} \textbf{29} (1965),
no.~4, 903--934 (Russian).
\href{https://www.mathnet.ru/eng/im3080}{MathNet record and original text}.
Correction, \emph{ibid.} \textbf{30} (1966), no.~3, 719--720;
\href{https://www.mathnet.ru/eng/im2856}{MathNet record and correction}.
\end{thebibliography}
==== END Simultaneous-primitive-roots-a-conditional-lower-bound-for-prime-bases//references.tex ====
==== BEGIN Simultaneous-primitive-roots-a-conditional-lower-bound-for-prime-bases//sections/00-abstract.tex ====
\begin{abstract}
For every fixed finite set of distinct positive primes, we prove that
at least $cx/(\log x)^2$ primes in $(x,2x)$ have every member of the set
as a primitive root, for some $c>0$ and all sufficiently large $x$.
The result assumes four explicitly stated analytic and sieve inputs
from the companion paper on primitive roots for admissible integer bases.
\end{abstract}
==== END Simultaneous-primitive-roots-a-conditional-lower-bound-for-prime-bases//sections/00-abstract.tex ====
==== BEGIN Simultaneous-primitive-roots-a-conditional-lower-bound-for-prime-bases//sections/01-introduction.tex ====
\section{Introduction}
For a prime \(p\) and an integer \(q\) not divisible by \(p\), let
\(\ord_p(q)\) be the multiplicative order of \(q\) in
\(\mathbb F_p^\times\). Thus \(q\) is a primitive root modulo \(p\) when
\(\ord_p(q)=p-1\). We study primes for which every member of a fixed finite
collection of positive prime bases is a primitive root.
The problem for one base is a case of Artin's primitive-root conjecture,
recorded by Hasse in 1927 \cite[Section~4.8]{HasseDiaries2012}.
Hooley proved the predicted asymptotic for one fixed integer under suitable
generalized Riemann hypotheses \cite{Hooley1967}.
Matthews obtained density results for simultaneous primitive roots under
generalized Riemann hypotheses \cite{Matthews1976}. Anwar and Pappalardi
later established an infinitude criterion under Schinzel's Hypothesis~H
\cite{AnwarPappalardi2017}.
Unconditional results allowing a finite choice of bases were obtained by Gupta and Murty
\cite{GuptaMurty1984} and by Heath-Brown \cite{HeathBrown1986}.
In particular, Heath-Brown's theorem implies that at least one member of
any triple of distinct positive primes is a primitive root modulo
infinitely many primes.
Our result is an implication from four statements taken from the companion
manuscript \cite{PrimitiveRoot2}: a uniform Hecke zero-free assertion
(Input~\ref{input:hecke}), a marked Type~II estimate
(Input~\ref{input:marked}), a block sieve (Input~\ref{input:block}), and a
rough-number density estimate (Input~\ref{input:density}). We reproduce
these statements in the precise forms used here. They are hypotheses of
the following theorem; their proofs in \cite{PrimitiveRoot2} and its cited
companion manuscripts are outside the scope of this paper.
\begin{theorem}\label{thm:main}
Assume Inputs~\ref{input:hecke}, \ref{input:marked},
\ref{input:block}, and~\ref{input:density}. Let \(k\ge1\) and let
\(q_1,\ldots,q_k\) be distinct positive primes. There are constants
\(c_{\mathbf q}>0\) and \(x_{\mathbf q}>1\) such that, for every real
\(x\ge x_{\mathbf q}\), at least
\[
c_{\mathbf q}\frac{x}{(\log x)^2}
\]
primes \(p\in(x,2x)\) satisfy
\[
p\nmid q_1\cdots q_k,
\qquad
\ord_p(q_i)=p-1\quad(1\le i\le k).
\]
\end{theorem}
We extend the construction of primes with controlled predecessors in
\cite[Section~12]{PrimitiveRoot2} to a fixed progression. The marked
Type~II estimate remains available after multiplication by a fixed
Dirichlet character, and Bombieri--Vinogradov supplies the required
distribution in that progression. A direct sieve for the final balanced
semiprimes keeps its constant independent of the later marking parameters.
The progression makes every prescribed base a quadratic nonresidue.
Uniform splitting estimates in the individual Kummer fields then remove
the remaining prime divisors of each order index. Section~2 gives this
reduction; the subsequent sections prove the two estimates it uses.
==== END Simultaneous-primitive-roots-a-conditional-lower-bound-for-prime-bases//sections/01-introduction.tex ====
==== BEGIN Simultaneous-primitive-roots-a-conditional-lower-bound-for-prime-bases//sections/02-reduction.tex ====
\section{Reduction to two estimates}
Write \(L=\log x\) and set \(\eta=10^{-6}\). For a prime \(p\nmid q\), write
\[
\iota_p(q)=\frac{p-1}{\ord_p(q)}
\]
for the order index of \(q\) modulo \(p\). The first proposition supplies
primes whose predecessors have one large prime factor and otherwise only
prime factors in a controlled interval.
\begin{proposition}[A prime reservoir in a fixed progression]
\label{prop:reservoir}
Assume Inputs~\ref{input:marked}, \ref{input:block},
and~\ref{input:density}. Let \(M\ge1\) be odd and squarefree, and let
\(a\) be a residue class modulo \(M\) with
\(\gcd(a(a-1),M)=1\). There are constants \(c_{M,a}>0\) and \(x_{M,a}>1\)
such that, for every real \(x\ge x_{M,a}\), at least
\(c_{M,a}x/L^2\) primes \(p\) satisfy
\[
xx^{0.9}\) is prime, and
\[
\exp(L^{0.1})<\ell<\exp(L^{0.3})
\qquad\text{for every prime }\ell\mid r.
\]
The case \(M=1\) is included.
\end{proposition}
The second proposition bounds the exceptional primes at which a controlled
prime factor can divide the index of a prescribed base.
\begin{proposition}[A uniform bound for order obstructions]
\label{prop:splitting}
Assume Input~\ref{input:hecke}. For each fixed positive prime \(q\), for all
sufficiently large real \(x\), and uniformly over primes \(\ell\) with
\(\exp(L^{0.1})<\ell<\exp(L^{0.3})\), one has
\[
\#\left\{\,p\text{ prime}:
\begin{array}{l}
x2,\\[6pt]
\displaystyle(-1)^{(p^2-1)/8}=-1,&q_i=2.
\end{cases}
\]
For a fixed one of the bases \(q=q_i\), the index \(\iota_p(q)\) is odd:
if it were even, then
\(\ord_p(q)\mid(p-1)/2\), contrary to Euler's criterion.
If \(\iota_p(q)>1\), it has an odd prime divisor \(\lambda\). Since
\(\iota_p(q)\mid4rQ\), either \(\lambda=Q\) or \(\lambda\mid r\), and
in either case
\begin{equation}\label{eq:index-obstruction}
q^{(p-1)/\lambda}\equiv1\pmod p.
\end{equation}
If \(Q\mid\iota_p(q)\), the integer \(j=(p-1)/Q\) is less than
\(2x^{0.1}\), and \(p\mid q^j-1\). The product of \(q^j-1\) over
\(1\le j\le2x^{0.1}\) has logarithm at most
\[
(\log q)\sum_{j\le2x^{0.1}}j\ll_q x^{0.2}.
\]
Each of its prime divisors in \((x,2x)\) contributes more than \(L\) to
this logarithm. Consequently the number of these exceptional primes is
\begin{equation}\label{eq:large-factor-exceptions}
O_q\!\left(\frac{x^{0.2}}{L}\right)
=o\!\left(\frac{x}{L^2}\right).
\end{equation}
In the other case, write \(\ell=\lambda\mid r\). The controlled interval
in Proposition~\ref{prop:reservoir} applies to \(\ell\). Since
\(\ell\mid p-1\), one has \(p\equiv1\pmod\ell\), while
Equation~\eqref{eq:index-obstruction} supplies the other congruence in
Proposition~\ref{prop:splitting}. Sum that proposition
over all such primes \(\ell\). The sum of \(1/(\ell(\ell-1))\) is at most
the telescoping sum over all integers exceeding \(\exp(L^{0.1})\), and
there are at most \(\exp(L^{0.3})\) possible \(\ell\). The number of
exceptions of this second kind is therefore
\begin{equation}\label{eq:controlled-factor-exceptions}
O_q\!\left(
\frac{x}{L}\exp(-L^{0.1})
+\exp(L^{0.3})x^{1-\eta}
\right)
=o\!\left(\frac{x}{L^2}\right).
\end{equation}
Here the two ratios to \(x/L^2\) are bounded by constant multiples of
\(L\exp(-L^{0.1})\) and
\(L^2\exp(L^{0.3}-\eta L)\), respectively, and both tend to zero.
There are only finitely many bases. The union over them of the exceptions
in Equations~\eqref{eq:large-factor-exceptions} and
\eqref{eq:controlled-factor-exceptions} is still \(o(x/L^2)\). Since
\(\#\mathcal R(x)\ge c_{M,a}x/L^2\), for every sufficiently large real
\(x\) at least \(c_{M,a}x/(2L^2)\) primes in \(\mathcal R(x)\) have
\(\iota_p(q_i)=1\) for every \(i\). These primes give the theorem.
\end{proof}
==== END Simultaneous-primitive-roots-a-conditional-lower-bound-for-prime-bases//sections/02-reduction.tex ====
==== BEGIN Simultaneous-primitive-roots-a-conditional-lower-bound-for-prime-bases//sections/03-splitting.tex ====
\section{A uniform splitting estimate}\label{sec:splitting}
The analytic input is the following assertion of
\cite[Theorem~1.2]{PrimitiveRoot2}. A finite-order Hecke character of a
number field \(F\) means a continuous finite-image character of
\(F^\times\backslash\mathbb A_F^\times\).
\begin{hypothesis}[Uniform Hecke zero-free assertion]\label{input:hecke}
Let \(F\) be a cyclotomic number field containing \(\mu_{12}\). For every
finite-order Hecke character \(\chi\) of \(F\), the meromorphic continuation
of \(L_F(s,\chi)\) has no zeros in
\[
\Re s>1-10^{-6}.
\]
\end{hypothesis}
The assertion permits the pole at \(s=1\) for the principal character.
We write \(\eta=10^{-6}\) throughout this section.
\begin{proof}[Proof of Proposition~\ref{prop:splitting}]
Fix the positive prime \(q\), put \(L=\log x\), and let \(\ell\) be a prime
in the range of the proposition. We take \(x\) large enough in terms of
\(q\) that \(\ell>q\) and \(\ell\ge5\). Set
\[
E_\ell=\mathbb Q(\mu_\ell),\qquad
K=K_{\ell,q}=E_\ell(q^{1/\ell}),\qquad
n=[K:\mathbb Q],\qquad D=|d_K|.
\]
The field \(K\) is the splitting field of \(T^\ell-q\), so it is Galois
over \(\mathbb Q\). The cyclotomic discriminant is
\(|d_{E_\ell}|=\ell^{\ell-2}\). Since \(q\ne\ell\), the prime \(q\) is
unramified in \(E_\ell\), and its valuation at each prime of \(E_\ell\)
above \(q\) is one. The polynomial \(T^\ell-q\) is Eisenstein at such a
prime. Consequently
\begin{equation}
[K:E_\ell]=\ell,\qquad n=\ell(\ell-1).
\label{eq:kummer-degree}
\end{equation}
The discriminant of the power basis generated by \(q^{1/\ell}\) is, up to
sign, \(\ell^\ell q^{\ell-1}\). The relative field discriminant therefore
divides \((\ell^\ell q^{\ell-1})\mathcal O_{E_\ell}\). The discriminant
tower formula gives
\begin{equation}
\begin{aligned}
D&\le \ell^{\ell(\ell-2)}
\bigl(\ell^\ell q^{\ell-1}\bigr)^{\ell-1}
=\ell^{\ell(2\ell-3)}q^{(\ell-1)^2},\\
\log D+n&\ll_q n\log(2\ell).
\end{aligned}
\label{eq:kummer-discriminant}
\end{equation}
The same divisibility and the cyclotomic discriminant show that rational
primes outside \(q\ell\) are unramified in \(K\).
We identify the two congruences in Proposition~\ref{prop:splitting} with
complete splitting. Let \(p\nmid q\ell\) be a prime, choose a prime
\(\mathfrak P\) of \(K\) above \(p\), and write
\(\sigma=\operatorname{Frob}_{\mathfrak P}\) for arithmetic Frobenius.
Its restriction to \(E_\ell\) sends \(\zeta_\ell\) to \(\zeta_\ell^p\),
so it is trivial exactly when \(p\equiv1\pmod\ell\). Under this condition,
put \(\alpha=q^{1/\ell}\). Since \(p\nmid q\), \(\alpha\) is a unit at
\(\mathfrak P\), and the defining congruence for arithmetic Frobenius gives
\[
\frac{\sigma(\alpha)}{\alpha}\in\mu_\ell,\qquad
\frac{\sigma(\alpha)}{\alpha}
\equiv\alpha^{p-1}=q^{(p-1)/\ell}\pmod{\mathfrak P}.
\]
Reduction on \(\mu_\ell\) is injective at \(p\ne\ell\). Hence
\(q^{(p-1)/\ell}\equiv1\pmod p\) is equivalent to \(\sigma\) fixing
\(\alpha\). Together the two congruences say that \(\sigma\) fixes both
\(E_\ell\) and \(\alpha\), and hence is the identity on \(K\). Because
\(K/\mathbb Q\) is Galois, this is precisely complete splitting of \(p\).
We transfer Input~\ref{input:hecke} to \(\zeta_K\). Put
\[
F_\ell=\mathbb Q(\mu_{12\ell}),\qquad
\widetilde K=KF_\ell=F_\ell(q^{1/\ell}).
\]
Because \(F_\ell\) contains \(\mu_\ell\), the extension
\(\widetilde K/F_\ell\) is Galois and the map
\(\sigma\mapsto\sigma(q^{1/\ell})/q^{1/\ell}\) embeds its Galois group
in \(\mu_\ell\). It is thus cyclic. Also \(F_\ell/\mathbb Q\) is abelian
Galois, so \(\widetilde K/K\) is Galois and restriction embeds its Galois
group in \(\Gal(F_\ell/\mathbb Q)\). It too is abelian. Abelian Artin
formalism \cite[Chapter~VII, Corollary~(10.5) and Theorem~(10.6)]{Neukirch1999},
with the one-dimensional Artin functions identified with their associated
primitive Hecke functions, gives
\begin{equation}
\begin{aligned}
\zeta_{\widetilde K}(s)
&=\prod_{\chi\in\widehat{\Gal(\widetilde K/F_\ell)}}L_{F_\ell}(s,\chi),\\
\frac{\zeta_{\widetilde K}(s)}{\zeta_K(s)}
&=\prod_{\substack{\chi\in\widehat{\Gal(\widetilde K/K)}\\\chi\ne1}}
L_K(s,\chi).
\end{aligned}
\label{eq:abelian-zeta-factorizations}
\end{equation}
These are identities of meromorphic functions, including the local factors
at ramified primes. The first product is zero-free on \(\Re s>1-\eta\)
by Input~\ref{input:hecke}, since \(F_\ell\) is cyclotomic and contains
\(\mu_{12}\); its principal factor has the permitted pole at one. Each
factor of the second product is an entire nonprincipal finite-order Hecke
\(L\)-function \cite{Tate1967}. A zero of \(\zeta_K\) in this half-plane away from one
would therefore be a zero of \(\zeta_{\widetilde K}\). Hence
\begin{equation}
\zeta_K(s)\ne0\qquad(\Re s>1-\eta,\ s\ne1).
\label{eq:kummer-zero-free}
\end{equation}
The common global half-plane in the input makes this width uniform in
\(\ell\), without requiring uniform onsets for the auxiliary estimates in
its proof. This is the abelian tower argument of
\cite[Section~9, Equation~(9.2)]{PrimitiveRoot2}, specialized to the
base \(q\).
To count completely split primes uniformly as the field varies, we use
the prime-ideal calculation from
\cite[Section~9, Equations~(9.5)--(9.9)]{PrimitiveRoot2}. Fix a nonnegative
\(\phi\in C_c^\infty((0,\infty))\) with \(\phi\ge1\) on \([1,2]\), and put
\[
\Phi(s)=\int_0^\infty\phi(t)t^{s-1}\dd t,\qquad
\Psi_{K'}(x)=\sum_{\mathfrak p}\sum_{m\ge1}
\log\Norm\mathfrak p\,
\phi\left(\frac{(\Norm\mathfrak p)^m}{x}\right)
\]
for a number field \(K'\). Suppose that \(K'\) has degree
\(n'=r_1+2r_2\), absolute discriminant \(D'\), and no zeta zero in
\(\Re s>1-\eta\). Its nontrivial zeros \(\rho\) lie in
\(0<\Re\rho<1\), and in fact \(\Re\rho\le1-\eta\). The Mellin transform
\(\Phi\) is entire and has arbitrary inverse-power decay in height on
fixed vertical strips. Mellin inversion of the logarithmic derivative,
followed by a shift from \(\Re s=2\) to \(\Re s=-1/2\), gives
\begin{equation}
\Psi_{K'}(x)=x\Phi(1)-\sum_\rho x^\rho\Phi(\rho)
-(r_1+r_2-1)\Phi(0)
+O_\phi\left(x^{-1/2}(\log D'+n')\right).
\label{eq:smooth-prime-ideal-formula}
\end{equation}
Here zeros are counted with multiplicity. The term at the origin comes
from the zero of \(\zeta_{K'}\) there of order \(r_1+r_2-1\); it is absent
when that order is zero. There are no other trivial zeros between the two
lines. To see the uniformity of the error, the functional equation gives
on \(\Re s=-1/2\)
\[
\left|\frac{\zeta_{K'}'}{\zeta_{K'}}(s)\right|
\ll \log D'+n'\log(2+|\Im s|).
\]
Indeed the Euler series at \(1-s\), with real part \(3/2\), is \(O(n')\),
and the logarithmic derivatives of the archimedean factors contribute
\(O(n'\log(2+|\Im s|))\). The rapid decay of \(\Phi\) gives the error in
Equation~\eqref{eq:smooth-prime-ideal-formula}. The usual rectangular
contours may be taken with horizontal edges away from zeros. The zero count
below supplies such edges, and the Hadamard partial-fraction formula bounds
the logarithmic derivative polynomially on them for each fixed \(K'\).
The decay of \(\Phi\) then makes their integrals vanish; the same zero count
makes the zero sum absolutely convergent.
For the required zero count, let \(N_{K'}(H)\) count nontrivial zeros with
\(|\Im\rho|\le H\), including multiplicity. The uniform formula of
\cite[Corollary~1.2, Equation~(1.3)]{HasanalizadeShenWong2022} is
\[
N_{K'}(H)=\frac H\pi
\log\left(D'\left(\frac{H}{2\pi e}\right)^{n'}\right)
+O\bigl(\log D'+n'\log H+n'\bigr)\qquad(H\ge1).
\]
For \(|T|\ge2\), the unit band is bounded by
\(N_{K'}(|T|+2)-N_{K'}(|T|-1)\). The main term changes by
\(O(\log D'+n'\log(2+|T|))\) over this interval, and both errors have the
same bound. For \(|T|<2\), use \(N_{K'}(3)\). Thus
\begin{equation}
\#\{\rho:T\le\Im\rhonL\)
to \(\Psi_K(x)\). Equations~\eqref{eq:kummer-degree} and
\eqref{eq:kummer-discriminant} therefore give
\begin{equation}
\begin{aligned}
&\#\{p\text{ prime}:x1\), and put \(P^-(1)=\infty\). An integer with
\(P^-(h)>W\) is called \(W\)-rough. Every parameter below, apart from an
explicitly displayed dependence on \(x\), is fixed before \(x\) grows.
For an integer \(K\ge1\) and fixed exponents
\(0.1x^{0.9}\) prime, so the complete group part supplies the factor \(r\)
in \(d-1=4rQ\).
The rough-factor density used below is, for \(w>\gamma>0\),
\begin{equation}\label{eq:rough-density}
D_\gamma(w)=\frac1w+
\sum_{j\ge2}\frac1{j!}
\int_{\substack{t_i\ge\gamma\ (1\le i\gamma>0\).
For completeness, in a term with \(j\ge2\) put \(s=w-j\gamma\). When
\(s\ge0\), the substitution \(t_i=\gamma+s y_i\) writes that term as
\[
\frac{s^{j-1}}{j!}
\int_{\substack{y_i\ge0\\\sum_{i\gamma>0\) the denominators stay bounded
away from zero. Extending this expression by zero for \(s<0\) proves
continuity also at \(w=j\gamma\). Local finiteness completes the claim.
All Dirichlet characters in the following input, including principal
characters, are extended by zero on nonunits. All frequencies are real.
Part (i) of the next input transfers scalar character cancellation to the
marked bilinear sum. Part (ii) supplies that cancellation for the
difference of the two roughness tests, with
\(D_\gamma(\log m/L)/(LV(W))\) as the comparison factor.
These are the coefficient criterion of
\cite[Lemma~10.2]{PrimitiveRoot2}, specialized to the marks above, and the
cancellation verified in the proof of
\cite[Proposition~10.3]{PrimitiveRoot2}.
\begin{hypothesis}[Marked Type II estimate]\label{input:marked}
\emph{(i) Coefficient criterion.}
Fix \(\delta,C,D_*>0\). Suppose \(H_m,H_n\ge x^\delta\) and
\(X=H_mH_n\asymp x\), with fixed comparison constants. Let
\(F:\mathbb Z_{>0}\to\mathbb C\) satisfy \(|F(h)|\le1\) and
\(F(ph)=F(h)\) for every group prime \(p\) and every \(h\ge1\).
Let \((\alpha_m)\) and \((\beta_n)\) be scalar sequences supported on
\(W\)-rough integers in \([H_m,2H_m]\) and \([H_n,2H_n]\), respectively,
with \(\alpha\) supported on an arbitrary interval
\(I\subseteq[H_m,2H_m]\), and with
\(|\alpha_m|,|\beta_n|\le L^C\). Suppose that for every fixed
\(A_0,B,A>0\), every Dirichlet character \(\chi\) modulo
\(k\le L^{A_0}\), and every \(|t|\le2XL^B\),
\begin{equation}\label{eq:marked-cancellation}
\left|\frac1{H_m}\sum_m\alpha_m\chi(m)m^{it}\right|\ll_A L^{-A},
\end{equation}
uniformly in the permitted interval, character, and frequency. The
constant and threshold in this hypothesis may depend on \(A_0,B,A\),
the fixed setup, and all further fixed parameters defining the
coefficients. Then
\begin{equation}\label{eq:marked-input-bound}
\left|\sum_{m,n}\alpha_m\beta_n F(mn-1)\mathcal W(mn-1)\right|
\ll XL^{-D_*}
\end{equation}
when \(K\) is sufficiently large in terms of the fixed data, independently
of the particular \(a_i\). Its implicit constant and threshold may depend
on all fixed data, including the common constants and thresholds in
\eqref{eq:marked-cancellation} and the \(a_i\). With those data fixed, the
bound is uniform over the permitted \(F\), dyads, intervals, and
coefficients.
\emph{(ii) Rough coefficient cancellation.}
Retain the marked data and the hypotheses on \(F,\beta,H_m,H_n\), without
assuming a previously chosen \(\alpha\). Fix \(0<\gammax^\gamma}
-\frac{D_\gamma(\log m/L)}{LV(W)}\one_{P^-(m)>W}\right)
\quad(m\in I),
\end{equation}
set equal to zero outside \(I\), satisfies
\eqref{eq:marked-cancellation} with \(\alpha^{(0)}\) in place of
\(\alpha\), for every fixed \(A_0,B,A>0\). This assertion is uniform in
the permitted dyads, interval, character, frequency \(t\), and \(v\).
Its constant and threshold may depend on the fixed parameters, including
\(\gamma,w_-,w_+\); in particular \(\gamma\) and \(w_--\gamma\) are
fixed before \(x\) grows.
\end{hypothesis}
Input~\ref{input:marked}(i) permits \(F\) to vary with \(x\), since the estimate is
uniform over all functions satisfying its two displayed conditions.
The next lemma inserts the fixed progression into the coefficient.
\begin{lemma}[A rough Type II estimate in a fixed progression]
\label{lem:twisted-rough-type-ii}
Fix \(\delta,C,D_*>0\) and \(0<\gammax^\gamma}
-\frac{D_\gamma(\log m/L)}{LV(W)}\one_{P^-(m)>W}\right),
\qquad |v|\le L^C,
\end{equation}
and set \(\alpha_m=0\) outside \(I\). Let \(\beta\) be supported on
\(W\)-rough integers in \([H_n,2H_n]\), with \(|\beta_n|\le L^C\). Then
\begin{equation}\label{eq:twisted-rough-type-ii}
\left|\sum_{m,n}\alpha_m\beta_nF(mn-1)\mathcal W(mn-1)\right|
\ll XL^{-D_*}
\end{equation}
when \(K\) is sufficiently large in terms of the fixed data, independently
of the particular \(a_i\). The estimate is uniform in the permitted
\(F\), dyads, intervals, \(\beta\), real \(v\), and characters \(\chi_0\).
The implicit constant and threshold may depend on all fixed data,
including \(M\) and the \(a_i\).
\end{lemma}
\begin{proof}
Let \(\alpha^{(0)}\) be the coefficient in
\eqref{eq:untwisted-rough-coefficient}. For a character \(\chi\) modulo
\(k\le L^{A_0}\), the product \(\theta=\chi_0\chi\), with the stated
zero extensions, is a character modulo \(h=\operatorname{lcm}(M,k)\).
Indeed it is a homomorphism on the units modulo \(h\), and it vanishes
exactly on the nonunits modulo \(h\). Since \(M\) is fixed,
\[
h\le ML^{A_0}\le L^{A_0+1}
\]
for sufficiently large \(x\). Input~\ref{input:marked}(ii), applied with
exponent \(A_0+1\), therefore gives
\[
\left|\frac1{H_m}\sum_m\alpha_m\chi(m)m^{it}\right|
=\left|\frac1{H_m}\sum_m\alpha_m^{(0)}\theta(m)m^{it}\right|
\ll_A L^{-A}
\qquad(|t|\le2XL^B).
\]
This includes imprimitive \(\theta\) and the case where \(\theta\) is
principal. It is uniform in \(\chi_0\), since all the product characters
belong to the same enlarged modulus range.
These scalar sums and their permitted test ranges do not depend on \(K\)
or the \(a_i\): \(W\) is defined from \(x\) alone. Fix their common
cancellation bounds using any one admissible auxiliary marked setup in
Input~\ref{input:marked}(ii). The same scalar bounds then apply unchanged when
the \(K\) and bands used in Input~\ref{input:marked}(i) are chosen.
For large \(x\), \(x^\gamma>W\), so both terms of \(\alpha\) are supported
on \(W\)-rough integers. The density is bounded on \([w_-,w_+]\), and
Mertens' theorem gives \(LV(W)\asymp L^{0.76}\). Hence \(\alpha\) is
bounded, and in particular \(|\alpha_m|\le L^C\) eventually; multiplication
by \(\chi_0\) does not increase its size. All the hypotheses of
Input~\ref{input:marked}(i) now hold, proving the lemma.
\end{proof}
The product progression will be imposed by the exact orthogonality identity
\begin{equation}\label{eq:product-progression}
\one_{mn\equiv a\pmod M}
=\frac1{\varphi(M)}\sum_{\chi_0\bmod M}
\overline{\chi_0(a)}\chi_0(m)\chi_0(n).
\end{equation}
If either factor is a nonunit modulo \(M\), both sides are zero because
\(a\) is a unit. For \(M=1\), the sum consists of the single trivial
character. This identity leaves \(F\) with its required invariance.
We now choose the invariant function to encode the desired predecessors.
Fix a smooth function \(\Psi\) compactly supported in \((1,2)\), with
\(0\le\Psi\le1\) and \(A_\Psi=\int_1^2\Psi(u)\dd u>0\). From now on take
\begin{equation}\label{eq:marked-family}
\begin{split}
F(h)&=\one_{\{h/h_{\mathcal P}=4Q\text{ for a prime }Q>x^{0.9}\}},
\qquad h\ge1,\\
w(d)&=\one_{d\equiv a\pmod M}\Psi(d/x)F(d-1)\mathcal W(d-1),
\qquad d\ge2.
\end{split}
\end{equation}
The quotient \(h/h_{\mathcal P}\) is unchanged by multiplication by any
group prime, even one already dividing \(h\), so this \(F\) is permitted
in Lemma~\ref{lem:twisted-rough-type-ii}. For large \(x\), every group
prime is coprime to \(2M\) and lies in
\((\exp(L^{0.1}),\exp(L^{0.3}))\), while every prime \(Q>x^{0.9}\) lies
outside the groups. Consequently, if \(w(d)>0\), there is a unique
representation
\begin{equation}\label{eq:marked-family-support}
\begin{gathered}
xx^{0.9}\text{ prime}.
\end{gathered}
\end{equation}
Moreover \(\mathcal W(d-1)=\mathcal W(r)\) and
\(r<(2x)/(4x^{0.9})\le x^{0.11}\). We also have \(0\le w(d)\le B_K\).
For \(r\in\mathcal G\), the progression in \eqref{eq:marked-family}
prescribes the reduced class
\[
b_r\equiv(a-1)(4r)^{-1}\pmod M
\]
for \(Q\). It is reduced by \eqref{eq:admissible-progression}; inverses
and classes for \(M=1\) have their trivial meaning. Define
\begin{equation}\label{eq:marked-mass-definitions}
\begin{split}
A_r&=\sum_{\substack{Q>x^{0.9}\text{ prime}\\Q\equiv b_r\pmod M}}
\Psi((4rQ+1)/x),\\
X_0&=\sum_{d\ge2}w(d)=\sum_{r\in\mathcal G}\mathcal W(r)A_r,\\
J_0&=\sum_{r\in\mathcal G}\frac{\mathcal W(r)}r.
\end{split}
\end{equation}
The equality for \(X_0\) follows from the uniqueness in
\eqref{eq:marked-family-support}. All these sums are nonnegative.
\begin{lemma}[Mass of the marked family]\label{lem:marked-mass}
For every fixed \(\vartheta>0\),
\begin{equation}\label{eq:marked-harmonic-bounds}
1\le J_0\ll_K1,\qquad
\sum_{\substack{r\in\mathcal G\\r>x^\vartheta}}
\frac{\mathcal W(r)}r
\ll_K\exp(-\vartheta L^{1-a_K}).
\end{equation}
For \(r\in\mathcal G\) with \(r\le x^{0.05}\), put
\[
I_r=\frac{x}{4r}\int_1^2
\frac{\Psi(u)}{\log((xu-1)/(4r))}\dd u.
\]
For every fixed \(A>0\), uniformly in these \(r\),
\begin{equation}\label{eq:progression-pnt}
A_r=\frac{I_r}{\varphi(M)}+O_{M,\Psi,A}\left(\frac xr L^{-A}\right),
\qquad I_r\asymp_\Psi\frac{x}{rL}.
\end{equation}
Furthermore,
\begin{equation}\label{eq:marked-mass-comparability}
X_0\asymp_{M,\Psi}\frac{xJ_0}{L}.
\end{equation}
The comparison constants here can be chosen independently of \(K\) and
the \(a_i\); the threshold for \(x\) may depend on them. For every fixed
\(\vartheta,A>0\),
\begin{equation}\label{eq:marked-tail}
\sum_{\substack{r\in\mathcal G\\r>x^\vartheta}}
\mathcal W(r)A_r=o(X_0L^{-A}).
\end{equation}
\end{lemma}
\begin{proof}
Put \(s=L^{-a_K}\). Every group prime satisfies \(p^s\le e^2\), and for
large \(x\), \(p^{s-1}\le e^2/p\le1/2\). The bound
\eqref{eq:mark-bound} and an Euler product including all prime powers give
\[
\sum_{r\in\mathcal G}\mathcal W(r)r^{s-1}
\le B_K\prod_{p\in\bigcup_i\mathcal P_i}(1-p^{s-1})^{-1}
\le B_K\exp\left(2e^2\sum_iV_i\right)\ll_K1.
\]
This bounds \(J_0\) above. Restricting \(J_0\) to
\(r=p_1\cdots p_K\), with one prime \(p_i\in\mathcal P_i\) to exponent
one, contributes exactly
\[
\prod_{i=1}^K\left(\frac1{V_i}\sum_{p_i\in\mathcal P_i}\frac1{p_i}\right)
=1.
\]
The disjoint groups make these products distinct. Finally, Rankin's
inequality gives
\[
\sum_{\substack{r\in\mathcal G\\r>x^\vartheta}}\frac{\mathcal W(r)}r
\le x^{-\vartheta s}\sum_{r\in\mathcal G}\mathcal W(r)r^{s-1}
\ll_K\exp(-\vartheta L^{1-a_K}),
\]
proving \eqref{eq:marked-harmonic-bounds}.
If \(r\le x^{0.05}\) and \(1\le u\le2\), then, for large \(x\),
\[
\frac{x^{0.95}}8\le\frac{xu-1}{4r}\le\frac x2,
\qquad
0.9L\le\log\left(\frac{xu-1}{4r}\right)\le L.
\]
In particular the cutoff \(Q>x^{0.9}\) is automatic on the smooth
support. The prime number theorem for the fixed modulus \(M\), uniformly
over its reduced classes and with arbitrary fixed logarithmic accuracy,
applied to \(t\mapsto\Psi((4rt+1)/x)\), proves the first assertion of
\eqref{eq:progression-pnt} by partial summation. Indeed this test is
supported on \(t\asymp x/r\), whose logarithm is comparable to \(L\), and
its total variation is \(\int|\Psi'(u)|\dd u\), independently of \(r\).
The continuous prime main term is exactly \(I_r/\varphi(M)\) after the
substitution \(u=(4rt+1)/x\). The displayed logarithmic bounds and
\(A_\Psi>0\) give the comparison for \(I_r\), with constants independent
of \(K\).
Write \(J_R=\sum_{r\in\mathcal G,\ r\le x^{0.05}}\mathcal W(r)/r\).
Taking \(A=3\) in \eqref{eq:progression-pnt} and summing with the marks
shows that \(\sum_{r\le x^{0.05}}\mathcal W(r)A_r\) is bounded above and
below by positive constants depending only on \(M,\Psi\) times
\(xJ_R/L\), once \(x\) is sufficiently large. The summed error is
\(O_{M,\Psi}(xJ_RL^{-3})\). For every \(r\), the smooth support forces
\(Qx^{0.05}}}\mathcal W(r)A_r
\ll_K x e^{-0.05L^{1-a_K}}.
\]
Since \(J_0\ge1\), after a threshold depending on \(K,a_i\) we have
\(J_R\ge J_0/2\), and the last sum is at most \(xJ_0/L\). Combining these
bounds proves \eqref{eq:marked-mass-comparability} with comparison
constants depending only on \(M,\Psi\). For general fixed \(\vartheta>0\),
the same bound for \(A_r\) gives a tail
\(O_K(xe^{-\vartheta L^{1-a_K}})\). Divide by
\(X_0L^{-A}\gg_{M,\Psi}xJ_0L^{-A-1}\) and use \(J_0\ge1\). Since
\(L^{1-a_K}\) dominates every fixed multiple of \(\log L\), the quotient
tends to zero. This proves \eqref{eq:marked-tail}.
\end{proof}
Fix \(0<\kappa<0.01\) and \(0<\epsilon<\kappa\), still before letting
\(x\) grow. Lemma~\ref{lem:marked-mass} gives \(X_0\asymp xJ_0/L\) for the
total weight, with \(J_0\) the harmonic sum that combines the
estimates for fixed \(r\). We now bound the summed divisibility remainders
for \(r\le x^\epsilon\); Equation~\eqref{eq:marked-tail} controls the
omitted tail. For odd squarefree \(\ell\) coprime to \(M\), define
\begin{equation}\label{eq:distribution-data}
\begin{split}
D_0&=x^{1/2-\kappa/2},\\
A_r(\ell)&=\sum_{\substack{Q>x^{0.9}\text{ prime}\\
Q\equiv b_r\pmod M\\\ell\mid4rQ+1}}
\Psi((4rQ+1)/x),\\
g_r(\ell)&=\frac{\one_{\gcd(\ell,r)=1}}{\varphi(\ell)},
\qquad R_r(\ell)=A_r(\ell)-g_r(\ell)A_r.
\end{split}
\end{equation}
On these squarefree integers \(g_r\) is multiplicative, and
\(A_r(1)=A_r\), \(g_r(1)=1\), \(R_r(1)=0\). If \(\gcd(\ell,r)>1\), both
terms in \(R_r(\ell)\) vanish. Otherwise the new condition prescribes
the reduced class \(Q\equiv-(4r)^{-1}\pmod\ell\). Together with
\(Q\equiv b_r\pmod M\), it gives one reduced class modulo \(M\ell\).
\begin{lemma}[Average distribution in the sieve moduli]
\label{lem:averaged-distribution}
For every fixed \(A>0\),
\begin{equation}\label{eq:averaged-distribution}
\sum_{\substack{r\in\mathcal G\\r\le x^\epsilon}}\mathcal W(r)
\sum_{\substack{\ell\le D_0\text{ odd and squarefree}\\
\gcd(\ell,M)=1}}
|R_r(\ell)|\ll xJ_0L^{-A}.
\end{equation}
The constant and threshold may depend on the fixed parameters.
\end{lemma}
\begin{proof}
Fix \(r\le x^\epsilon\). Since \(\epsilon<0.01<0.05\), the cutoff in
\(A_r(\ell)\) is automatic on its smooth support, as in
\eqref{eq:progression-pnt}. Put \(\delta_0=(\kappa-\epsilon)/4>0\).
For every \(t\asymp x/r\) on that support, all the distinct moduli
\(M\ell\), \(\ell\le D_0\), satisfy
\[
M\ell\le t^{1/2-\delta_0}
\]
for sufficiently large \(x\). In fact, the exponent margin at the
smallest possible scale is
\[
(1-\epsilon)(1/2-\delta_0)-(1/2-\kappa/2)
=\frac{(\kappa-\epsilon)(1+\epsilon)}4>0,
\]
which absorbs the fixed factor \(M\) and the comparison constants.
The Bombieri--Vinogradov theorem \cite{Bombieri1965,Vinogradov1965}, restricted to this
fixed-power sublevel, gives for every fixed \(A'>0\) an
\(O(t(\log t)^{-A'})\) sum of absolute errors in prime counts up to \(t\),
using the maximum over reduced classes and main term
\(\operatorname{Li}(t)/\varphi(M\ell)\). This follows from its usual
level \(t^{1/2}(\log t)^{-B}\), since \(\delta_0\) is fixed and positive.
Integrate those errors against the derivative of
\(\Psi((4rt+1)/x)\). Its total absolute integral is bounded in terms of
\(\Psi\), and \(t\asymp x/r\), \(\log t\asymp L\) throughout. The triangle
inequality under the integral thus gives
\[
\sum_{\substack{\ell\le D_0\text{ odd and squarefree}\\
\gcd(\ell,M)=1}}
\left|A_r(\ell)-g_r(\ell)\frac{I_r}{\varphi(M)}\right|
\ll \frac xr L^{-A'}.
\]
For \(\gcd(\ell,r)=1\), the main term uses
\(\varphi(M\ell)=\varphi(M)\varphi(\ell)\); the other terms are zero.
The maximum over reduced classes in the theorem makes the bound uniform
in the class depending on \(r,\ell\).
To replace \(I_r/\varphi(M)\) by \(A_r\), observe that
\[
\sum_{\substack{\ell\le D_0\\\ell\text{ squarefree}}}\frac1{\varphi(\ell)}
\le\prod_{p\le D_0}\left(1+\frac1{p-1}\right)
=V(D_0)^{-1}\ll L.
\]
Equation~\eqref{eq:progression-pnt}, with its logarithmic accuracy
increased by two, absorbs this factor. Taking \(A'\) sufficiently large
therefore proves \(\sum_\ell|R_r(\ell)|\ll(x/r)L^{-A}\), uniformly in
the present \(r\). Multiplying by \(\mathcal W(r)\), summing, and using
\(\sum_{r\le x^\epsilon}\mathcal W(r)/r\le J_0\) proves the lemma.
\end{proof}
The sieve products for this progression are
\begin{equation}\label{eq:progression-sieve-products}
\begin{split}
V_M(y)&=\prod_{\substack{22}\left(1-\frac1{(p-1)^2}\right)
\prod_{p\mid M}\left(1-\frac1{p-1}\right)^{-1}>0.
\end{split}
\end{equation}
The infinite product is positive because its factors are positive and
their deviations from one have a convergent sum. At every odd prime,
\[
\frac{1-1/(p-1)}{1-1/p}=1-\frac1{(p-1)^2}.
\]
The factor at \(2\) and the omitted factors at primes dividing \(M\)
therefore give, with Mertens' theorem,
\begin{equation}\label{eq:progression-product-asymptotic}
\frac{V_M(y)}{V(y)}\longrightarrow\mathfrak S_M,
\qquad V(y)\sim\frac{e^{-\gamma_E}}{\log y}
\quad(y\to\infty),
\end{equation}
where \(\gamma_E\) is Euler's constant.
Finally, for every \(r\in\mathcal G\) with \(r\le x^{0.11}\), its distinct
prime divisors exceed \(\exp(L^{0.1})\). Their number is at most
\(0.11L^{0.9}\), and hence
\begin{equation}\label{eq:group-reciprocal-divisors}
\sum_{p\mid r}\frac1p\le0.11L^{0.9}e^{-L^{0.1}}=o(1).
\end{equation}
For these primes, \(\log(1-1/(p-1))^{-1}\ll1/p\) with an absolute
constant. It follows uniformly for all such \(r\) and all \(y\ge1\) that
\begin{equation}\label{eq:missing-group-factors}
\begin{split}
\prod_{\substack{2x^{0.9}$ is prime, and it lies in the reduced
class $a$ modulo $M$. By Lemma~\ref{lem:marked-mass} and $w(d)\le B_K$,
it is enough to prove that the prime weight is $\gg X_0/L$.
The two sieve inputs needed below are recorded explicitly.
\begin{hypothesis}[Block sieve]\label{input:block}
Let $z>1$, let $\mathcal A$ be a finite set with nonnegative weights
$\lambda_\nu$ for $\nu\in\mathcal A$, and let $\mathcal D$ be a set of
designated primes at most $z$. For each $p\in\mathcal D$ let
$\mathcal A_p\subseteq\mathcal A$ be its bad condition. For a squarefree
product $\ell$ of primes in $\mathcal D$, put
$\mathcal A_\ell=\bigcap_{p\mid\ell}\mathcal A_p$, with
$\mathcal A_1=\mathcal A$. Suppose that, including for $\ell=1$,
\[
\sum_{\nu\in\mathcal A_\ell}\lambda_\nu=Yg(\ell)+R(\ell),
\qquad Y\ge0,
\]
where $g$ is multiplicative on these squarefree products and $g(1)=1$.
Extend $g(p)$ by zero at primes outside $\mathcal D$. Suppose that for
fixed constants $\eta_0>0$ and $C_0<\infty$,
\[
0\le g(p)\le1-\eta_0,
\qquad
\sum_{u1).
\]
Write
\[
S=\sum_{\nu\in\mathcal A\setminus\bigcup_{p\in\mathcal D}\mathcal A_p}
\lambda_\nu.
\]
For every sufficiently large even integer $H$, the weight avoiding all
the bad conditions is
\begin{equation}\label{eq:block-sieve}
\begin{split}
S={}&Y\prod_{p\le z}(1-g(p))\bigl(1+O_{\eta_0,C_0}(e^{-H})\bigr)\\
&+O_{\eta_0,C_0}\left(
\sum_{\substack{\ell\le z^{4H+2}\\
\ell\ \mathrm{squarefree}\\
p\mid\ell\Rightarrow p\in\mathcal D}}
|R(\ell)|\right).
\end{split}
\end{equation}
The threshold for $H$ and the implied constants depend
only on $\eta_0,C_0$. In particular, the constants are uniform when
$H$ grows, and the remainder sum has no additional coefficient or
multiplicity factor. For $H=2$ there is the upper bound
\begin{equation}\label{eq:block-sieve-upper}
S\ll_{\eta_0,C_0}
Y\prod_{p\le z}(1-g(p))+
\sum_{\substack{\ell\le z^{10}\\
\ell\ \mathrm{squarefree}\\
p\mid\ell\Rightarrow p\in\mathcal D}}
|R(\ell)|.
\end{equation}
Both remainder sums include only squarefree products of the designated
primes, with $\ell=1$ included. This is the form stated in
\cite[Lemma~11.1, Equations~(11.2)--(11.3)]{PrimitiveRoot2}.
\end{hypothesis}
\begin{hypothesis}[Rough-number density]\label{input:density}
For the density $D_\gamma(w)$ defined in Section~\ref{sec:marked-family},
there are absolute constants $b_d,c_d,C_d>0$ such that, for every fixed
$0z$ is equivalent
to excluding divisibility by the odd primes $p\le z$ with $p\nmid M$.
Fix a group integer $r\le x^\epsilon$. Apply Input~\ref{input:block}
to the primes $Q$ and their weights in $A_r$, with the bad condition
$p\mid4rQ+1$ at each odd $p\le x^b$ not dividing $M$. The intersection
counts are $A_rg_r(\ell)+R_r(\ell)$ by
\eqref{eq:distribution-data}. Compact support of $\Psi$ makes this
a finite weighted family after removing zero weights. The densities
on designated primes
obey
\[
0\le g_r(p)\le\frac1{p-1}\le\frac12.
\]
Their sums on $(u,u^2]$ have an absolute bound, since
$1/(p-1)\le2/p$ for odd $p$ and the prime harmonic sums on these
intervals are bounded. Thus the local constants of the block sieve
are absolute, independently of $M,r,K$.
Use the even depth
\[
H_b=2\left\lfloor\frac1{40b}\right\rfloor.
\]
For sufficiently small $b$ it is an admissible depth in
Input~\ref{input:block}, and
\begin{equation}\label{eq:initial-sieve-level}
H_b\ge\frac1{20b}-2,
\qquad
b(4H_b+2)\le\frac15+2b<0.22<\frac12-\frac\kappa2.
\end{equation}
In particular, every remainder modulus is at most
$x^{b(4H_b+2)}\le D_0$. Sum the resulting lower bound with the
nonnegative marks $\mathcal W(r)$. By
\eqref{eq:averaged-distribution} the sum of the remainders is
$O(xJ_0L^{-A})$ for any fixed $A$; taking $A>2$ makes it $o(X_0/L)$
by \eqref{eq:marked-mass-comparability}. The tail with
$r>x^\epsilon$ is negligible to this accuracy by
\eqref{eq:marked-tail}. The uniform formula for $\Pi_r(x^b)$ and
\eqref{eq:progression-product-asymptotic} give
\[
\Pi_r(x^b)\sim V_M(x^b)
\sim\frac{\mathfrak S_M e^{-\gamma_E}}{bL}.
\]
Finally, \eqref{eq:initial-sieve-level} gives
$e^{-H_b}\le e^2e^{-1/(20b)}$. We have proved
\begin{equation}\label{eq:initial-rough-mass}
\sum_{\substack{d\ge2\\P^-(d)>x^b}}w(d)
\ge\frac{\mathfrak S_M X_0}{L}
\left\{\frac{e^{-\gamma_E}}{b}
\bigl(1-C_se^{-c_s/b}\bigr)+o(1)\right\},
\end{equation}
where $C_s,c_s>0$ are absolute; for example, $c_s=1/20$ is permitted
after enlarging $C_s$. The implied threshold may depend on all fixed
parameters.
For fixed $b\le\gamma<\gamma'\le1/2-\kappa$, let
\[
\mathcal N_{\gamma,\gamma'}
=\{n\text{ prime}:x^\gammaW}}w(d)
=X_0V_M(W)\left\{\log\frac{\gamma'}\gamma+o(1)\right\}.
\end{equation}
For large $x$, every such $n$ exceeds $W$, every prime dividing $M$,
and every group prime. Indeed $x^b$ eventually exceeds each of these
quantities. In particular, $n\nmid r$ for every group integer $r$.
At fixed $r\le x^\epsilon$, condition the family of $A_r$ on
$n\mid4rQ+1$ and designate the odd primes $p\le W$ not dividing $M$.
For a squarefree product $\ell$ of these primes, including $\ell=1$,
\begin{equation}\label{eq:conditioned-intersections}
A_r(n\ell)=\frac{A_r}{n-1}g_r(\ell)+R_r(n\ell).
\end{equation}
This follows from $n\nmid Mr\ell$ and multiplicativity of $g_r$.
Thus the block sieve applies with $Y=A_r/(n-1)$, the same local
densities, and remainder $R_r(n\ell)$. Its remainder at $\ell=1$ is
$R_r(n)$, which is explicitly allowed by Input~\ref{input:block}.
Take $H_W=2\lceil(\log L)^2\rceil$. It is an admissible growing even
depth, its relative error tends to zero, and
\[
W^{4H_W+2}
=\exp\bigl(O(L^{0.24}(\log L)^2)\bigr)=x^{o(1)}.
\]
Every remainder modulus in \eqref{eq:conditioned-intersections}
therefore satisfies
\[
n\ell\le x^{1/2-\kappa+o(1)}2$ makes this
$o(X_0V_M(W))$, because
$X_0V_M(W)\asymp_{M,\Psi}xJ_0L^{-1.24}$.
The main sieve product is $\Pi_r(W)=V_M(W)(1+o(1))$ uniformly.
Also the prime harmonic estimate gives
\[
\sum_{n\in\mathcal N_{\gamma,\gamma'}}\frac1{n-1}
=\log\frac{\gamma'}\gamma+o(1).
\]
Here replacing $1/n$ by $1/(n-1)$ costs $o(1)$, and Mertens' theorem
evaluates the sum of $1/n$. These facts give the main term in
\eqref{eq:conditioned-bin-count} for $r\le x^\epsilon$. For the
omitted tail, a supported $d<2x$ has at most
$\log(2x)/(bL)=O_b(1)$ distinct prime divisors above $x^b$.
Multiplying the tail bound \eqref{eq:marked-tail} by this fixed
factor still gives $o(X_0V_M(W))$. This proves
\eqref{eq:conditioned-bin-count}.
\subsection{Removing the least prime factor}
If the least prime factor of a supported composite $d$ belongs to
the bin $(x^\gamma,x^{\gamma'}]$, write it as $n$ and put $m=d/n$.
Then $P^-(m)\ge n>x^\gamma$, also when $n^2\mid d$. The weight of
these composites is consequently at most
\[
T_{\gamma,\gamma'}=
\sum_{n\in\mathcal N_{\gamma,\gamma'}}
\ \sum_{\substack{m\ge1\\P^-(m)>x^\gamma}}w(mn).
\]
Lemma~\ref{lem:twisted-rough-type-ii} gives the replacement
\begin{equation}\label{eq:rough-replacement}
T_{\gamma,\gamma'}=
\sum_{n\in\mathcal N_{\gamma,\gamma'}}
\ \sum_{\substack{m\ge1\\P^-(m)>W\\xL}|\widehat\psi(v)|\dd v\ll_{A,\Psi}L^{-A}
\end{gathered}
\end{equation}
for every fixed $A>0$. Insert the character expansion
\eqref{eq:product-progression} to separate the progression condition.
In each retained box apply Lemma~\ref{lem:twisted-rough-type-ii}
with
\[
\delta=b/2,\qquad C=2,\qquad D_*=6,
\]
and with the exponent interval above. For $|v|\le L$ its first
coefficient is the difference of the two tests in
\eqref{eq:rough-replacement}, multiplied by $\chi_0(m)m^{iv}$.
The second coefficient is
$\chi_0(n)n^{iv}\one_{n\text{ prime}}$, restricted to its bin and
dyad. It is bounded by one and is supported on $W$-rough integers,
since $n>W$. The function $F$ is exactly the invariant function
in \eqref{eq:marked-family}. The product condition remains in
$\Psi$, so neither coefficient has been restricted by a condition
on both factors. The lemma applies for sufficiently large $K$
depending on these fixed parameters, independently of the band
exponents. Its bound \eqref{eq:twisted-rough-type-ii} is
$O(xL^{-6})$ per box; integration over $|v|\le L$ costs only the
bounded $L^1$ norm in \eqref{eq:reservoir-fourier-separation}.
For $|v|>L$, the density is bounded on the fixed exponent interval
and $LV(W)\asymp L^{0.76}$. Thus the two discrepancy coefficients
are bounded for large $x$. The marked weight is bounded by $B_K$ by
\eqref{eq:mark-bound},
and a box has $O(H_mH_n)=O(x)$ integer pairs. The last bound in
\eqref{eq:reservoir-fourier-separation}, with sufficiently large
fixed $A$, makes this tail $O_{K,\Psi}(xL^{-6})$ per box as well. The
$O(L)$ boxes therefore contribute $O(xL^{-5})=o(X_0/L)$, with the
constant allowed to depend on all fixed parameters, using
\eqref{eq:marked-mass-comparability} and $J_0\ge1$. This proves
\eqref{eq:rough-replacement}. All occurrences of the density in
that equation are in its domain, by \eqref{eq:bin-exponent-range}.
By the joint continuity established in Section~\ref{sec:marked-family},
\eqref{eq:bin-exponent-range} implies, uniformly on the support in
\eqref{eq:rough-replacement},
\[
D_\gamma(\log m/L)
\le\sup_{\gamma\le t\le\gamma'}D_\gamma(1-t)+o(1).
\]
Since $n>W$, the condition $P^-(m)>W$ is equivalent to
$P^-(mn)>W$. For a fixed divisor $n\mid d$ the factor $m=d/n$ is
unique. We may therefore apply \eqref{eq:conditioned-bin-count}
after changing variables $d=mn$. Together with
$V_M(W)/V(W)\to\mathfrak S_M$, this gives
\begin{equation}\label{eq:least-factor-bin-upper}
T_{\gamma,\gamma'}\le\frac{\mathfrak S_M X_0}{L}
\left\{
\left(\sup_{\gamma\le t\le\gamma'}D_\gamma(1-t)\right)
\log\frac{\gamma'}\gamma+o(1)
\right\}.
\end{equation}
Choose a finite partition
$b=\gamma_0<\gamma_1<\cdots<\gamma_J=1/2-\kappa$.
The set of pairs $(\gamma,1-t)$ with
$b\le\gamma\le t\le1/2-\kappa$ is a compact subset of
$\{(\gamma,w):w>\gamma>0\}$, since $1-t-\gamma\ge2\kappa$.
Joint continuity of the density is uniform there. For every $t$ in a bin,
the bin supremum in \eqref{eq:least-factor-partition}, whose first argument
is fixed at the left endpoint, differs from $D_t(1-t)$ by a quantity
tending uniformly to zero with the mesh. The measure $\dd t/t$ has finite total mass
$\log((1/2-\kappa)/b)$. We may consequently choose a sufficiently
fine fixed partition for which
\begin{equation}\label{eq:least-factor-partition}
\begin{split}
\sum_{j=1}^J
\left(\sup_{\gamma_{j-1}\le t\le\gamma_j}
D_{\gamma_{j-1}}(1-t)\right)
\log\frac{\gamma_j}{\gamma_{j-1}}
&\le\int_b^{1/2-\kappa}D_t(1-t)\frac{\dd t}{t}+\frac18\\
&\le D_b(1)-1+\frac18.
\end{split}
\end{equation}
The last inequality uses nonnegativity of $D_t$ and
Input~\ref{input:density}.
Put
\[
U_\kappa=\sum_{\substack{d\ge2\\P^-(d)>x^{1/2-\kappa}}}w(d).
\]
Every supported integer counted in \eqref{eq:initial-rough-mass}
but not in the sum defining $U_\kappa$ is composite: it exceeds $x$,
whereas its
least prime factor is at most $x^{1/2-\kappa}$. That factor lies
in one of the bins of our partition. Subtracting
\eqref{eq:least-factor-bin-upper} for these finitely many bins from
\eqref{eq:initial-rough-mass}, and using
\eqref{eq:least-factor-partition} and
\eqref{eq:density-integral-bound}, yields
\begin{equation}\label{eq:post-bin-mass}
\frac{LU_\kappa}{\mathfrak S_M X_0}
\ge1-\frac18-\frac{e^{-\gamma_E}}{b}
\left(C_se^{-c_s/b}+C_de^{-c_d/b}\right)+o(1).
\end{equation}
Here the lower bound uses
$\frac{e^{-\gamma_E}}b(1-C_se^{-c_s/b})-
(D_b(1)-1+1/8)$; the stated upper bound on $D_b(1)$ gives precisely
the two exponential losses in \eqref{eq:post-bin-mass}.
\subsection{Balanced composites and the parameter choices}
It remains to bound the composite part of $U_\kappa$. The final choice of
$\kappa$ precedes those of $b$ and $K$, so the coefficient in this bound
must be independent of all three. We achieve this by sieving pairs for
each fixed $r$ before summing the marks. Such a composite has exactly two
prime factors counted with multiplicity.
Indeed three factors, each greater than $x^{1/2-\kappa}$, would have
product at least $x^{3/2-3\kappa}>2x$ for large $x$. Write the two
factors as $m,n$. Each is greater than $x^{1/2-\kappa}$ and less
than $2x^{1/2+\kappa}$, so for large $x$ both belong to
\[
[x^{1/2-2\kappa},x^{1/2+2\kappa}].
\]
Use ordered pairs $(m,n)$ to bound their weight; this counts a square
once and may count other semiprimes twice, which is harmless for an
upper bound.
Cover these pairs by disjoint dyadic boxes
$[M_1,2M_1)\times[M_2,2M_2)$ meeting the displayed factor range and
$xx^{1/2-2\kappa}>z$ and
$Q=(mn-1)/(4r)>x^{0.9}>z$ are primes. Thus $mn\equiv0\pmod j$
would force $j$ to be one of the prime factors $m,n$, and, since
$j\nmid4r$, $mn\equiv1\pmod j$ would force $j=Q$. Both are
impossible. We discard the progression condition modulo $M$ to
obtain an upper bound.
Use as a finite unweighted family all integer pairs in the box
satisfying the first congruence in
\eqref{eq:balanced-necessary-conditions}, and designate its odd
primes $j\le z$ not dividing $r$, with bad condition
$mn\equiv0\text{ or }1\pmod j$. All sums and products indexed by
$j$ in this balanced sieve are over primes. For a squarefree product
$\ell$ of these primes set
\[
\rho(\ell)=\prod_{j\mid\ell}(3j-2),\qquad \rho(1)=1.
\]
There are $\varphi(k_r)$ residue pairs modulo $k_r$ with product
one. At $j$ there are $2j-1$ residue pairs with product zero and
$j-1$ with product one, and these two sets are disjoint. Since
$(k_r,\ell)=1$, the Chinese remainder theorem gives exactly
$\varphi(k_r)\rho(\ell)$ residue pairs modulo $k_r\ell$ satisfying
the base congruence and all the designated bad conditions at
primes dividing $\ell$.
Each residue pair modulo $k_r\ell$ occurs in the box
\[
\frac{M_1M_2}{(k_r\ell)^2}
+O\left(\frac{M_1+M_2}{k_r\ell}+1\right)
\]
times. Thus its intersection count has the form $Y_rg(\ell)+R(\ell)$,
where
\begin{equation}\label{eq:balanced-sieve-data}
Y_r=\frac{M_1M_2\varphi(k_r)}{k_r^2},
\qquad g(\ell)=\frac{\rho(\ell)}{\ell^2}.
\end{equation}
\begin{equation}\label{eq:balanced-lattice-error}
\begin{split}
|R(\ell)|
&\ll\varphi(k_r)\rho(\ell)
\left(\frac{M_1+M_2}{k_r\ell}+1\right)\\
&\ll (M_1+M_2)\ell+k_r\ell^2.
\end{split}
\end{equation}
The last inequality uses $\varphi(k_r)\le k_r$ and
$\rho(\ell)\le\ell^2$, because $3j-2\le j^2$ for every odd prime
$j$. This count also includes $\ell=1$, so its boundary error for
the base family is included in $R(1)$.
For a designated prime,
$g(j)=(3j-2)/j^2\le7/9$ and $g(j)\le3/j$. Thus the local gap and
block sums in Input~\ref{input:block} again have absolute constants.
Its $H=2$ upper bound uses only $\ell\le z^{10}=x^{0.02}$.
Summing \eqref{eq:balanced-lattice-error} over this range, bounding
the designated squarefree products by all positive integers, gives
\begin{align}
\sum_{\ell\le x^{0.02}}|R(\ell)|
&\ll (M_1+M_2)\sum_{\ell\le x^{0.02}}\ell
+k_r\sum_{\ell\le x^{0.02}}\ell^2\notag\\
&\ll x^{0.54+0.04}+r x^{0.06}
\ll x^{0.58}+x^{0.17}.\label{eq:balanced-remainder-sum}
\end{align}
The notation in the left-hand sum restricts to the designated
products for which $R$ is defined. The exponent margins compared
with $x/r$ are uniform throughout $r\le x^{0.11}$:
\[
\frac{x^{0.58}}{x/r}=r x^{-0.42}\le x^{-0.31},
\qquad
\frac{r x^{0.06}}{x/r}=r^2x^{-0.94}\le x^{-0.72}.
\]
Both fixed power savings absorb $L^4$. It follows that
\eqref{eq:balanced-remainder-sum} is $o((x/r)L^{-4})$ uniformly in
every supported $r$, with no dependence on $K$ in this assertion.
For the main product the exact local identity is
\[
1-g(j)=\left(1-\frac1j\right)^3
\left(1-\frac1{(j-1)^2}\right).
\]
The last factor is at most one. The missing factor at two costs
only the factor $8$ when compared with $V(z)^3$. The missing
factors at primes dividing $r$ cost at most
\[
\prod_{j\mid r}\left(1-\frac1j\right)^{-3}
=\exp\left(O\left(\sum_{j\mid r}\frac1j\right)\right)=1+o(1)
\]
uniformly, since on the full support
$\sum_{j\mid r}j^{-1}\le0.11L^{0.9}e^{-L^{0.1}}$ by
\eqref{eq:group-reciprocal-divisors}. Consequently
\begin{equation}\label{eq:balanced-sieve-product}
\prod_{\substack{j\le z\\j\ \mathrm{odd\ prime}\\j\nmid r}}
(1-g(j))\ll V(z)^3\ll L^{-3},
\end{equation}
with an absolute constant. The numerical exponent $0.002$ is
fixed, so the last estimate follows from Mertens' theorem with a
fixed constant.
By \eqref{eq:balanced-box-range}, $Y_r\ll x/r$. Combining
\eqref{eq:block-sieve-upper}, \eqref{eq:balanced-remainder-sum},
and \eqref{eq:balanced-sieve-product} bounds the number of prime
pairs in this box contributing through $r$ by
\[
O\bigl((x/r)L^{-3}\bigr),
\]
uniformly for all supported $r$. Its constant is absolute; the
power-saving error above has been absorbed for sufficiently large
$x$. The smooth weight is at most one. Multiplying by
$\mathcal W(r)$ and summing over the full support of $r$, without
discarding a tail, therefore bounds the weighted prime pairs in
one box by $C_{\mathrm{pair}}xJ_0/L^3$ with an absolute
$C_{\mathrm{pair}}$. The unique
representation of the support justifies this assignment of each
weighted pair to its group integer.
Let $\mathcal C_\kappa$ denote the weight of the composites counted by
$U_\kappa$. Summing the last estimate over the boxes gives
\begin{equation}\label{eq:balanced-composite-mass}
\mathcal C_\kappa
\le C_2(\kappa+L^{-1})\frac{X_0}{L}.
\end{equation}
Here $C_2$ depends only on $M,\Psi$ and the fixed dyadic comparison
conventions, and is independent of $\kappa,b,K$. Indeed the
preceding bound before converting the normalization is
$C_{\mathrm{pair}}C_{\mathrm{dyad}}(\kappa L+1)xJ_0/L^3$.
The lower comparison in \eqref{eq:marked-mass-comparability} is
$X_0\ge c_{M,\Psi}xJ_0/L$ for large $x$, with
$c_{M,\Psi}>0$ independent of $K$. We may take
$C_2=C_{\mathrm{pair}}C_{\mathrm{dyad}}/c_{M,\Psi}$. The threshold for $x$ may
depend on all the fixed parameters; this does not change the stated
independence of the constant.
Every supported prime is counted by $U_\kappa$. Subtracting its
composite part using \eqref{eq:post-bin-mass} and
\eqref{eq:balanced-composite-mass} yields
\begin{equation}\label{eq:final-prime-mass}
\begin{split}
\frac{L}{\mathfrak S_M X_0}\sum_{d\text{ prime}}w(d)
&\ge1-\frac18-\frac{e^{-\gamma_E}}{b}
\left(C_se^{-c_s/b}+C_de^{-c_d/b}\right)\\
&\quad-\frac{C_2}{\mathfrak S_M}(\kappa+L^{-1})+o(1).
\end{split}
\end{equation}
We now choose the parameters. With $M$ fixed, the function $\Psi$ and
the dyadic convention have already fixed $C_2$, while
$\mathfrak S_M>0$ depends only on $M$. Choose
$0<\kappa<0.01$ so small that
$C_2\kappa/\mathfrak S_M<1/8$, and then choose
$0<\epsilon<\kappa$. Next choose fixed $b>0$ sufficiently small
for Inputs~\ref{input:block} and~\ref{input:density}, for
\eqref{eq:initial-sieve-level}, and so that
\[
\frac{e^{-\gamma_E}}{b}
\left(C_se^{-c_s/b}+C_de^{-c_d/b}\right)<\frac18.
\]
This is possible because $b^{-1}e^{-c/b}\to0$ as $b\downarrow0$
for each $c>0$. Fix a finite partition satisfying
\eqref{eq:least-factor-partition}. For its finitely many left
endpoints $\gamma$, use the fixed Type II choices
$\delta=b/2$, $C=2$, $D_*=6$ and the exponent interval already
specified. Now choose $K$ sufficiently large for
Lemma~\ref{lem:twisted-rough-type-ii} in every one of these
applications. The lower bound on $K$ is independent of the band
exponents, so choose any fixed
$0.1