\documentclass[11pt]{amsart} \usepackage[T1]{fontenc} \usepackage{lmodern} \usepackage{amsmath,amssymb,amsthm,mathtools} \usepackage[letterpaper,margin=1.05in]{geometry} \usepackage{microtype} \usepackage{enumitem} \usepackage[dvipsnames]{xcolor} \usepackage[colorlinks=true,linkcolor=MidnightBlue,citecolor=MidnightBlue, urlcolor=MidnightBlue,bookmarksnumbered=true]{hyperref} \numberwithin{equation}{section} \newtheorem{theorem}{Theorem}[section] \newtheorem{proposition}[theorem]{Proposition} \newtheorem{lemma}[theorem]{Lemma} \newtheorem{corollary}[theorem]{Corollary} \theoremstyle{definition} \newtheorem{definition}[theorem]{Definition} \theoremstyle{remark} \newtheorem{remark}[theorem]{Remark} \newtheorem{example}[theorem]{Example} \newcommand{\Q}{\mathbb Q} \newcommand{\Z}{\mathbb Z} \newcommand{\C}{\mathbb C} \newcommand{\N}{\mathbb N} \newcommand{\F}{\mathbb F} \newcommand{\eps}{\varepsilon} \newcommand{\ind}{\mathbf 1} \DeclareMathOperator{\Gal}{Gal} \DeclareMathOperator{\ord}{ord} \DeclareMathOperator{\Nm}{N} \DeclareMathOperator{\Frob}{Frob} \DeclareMathOperator{\rad}{rad} \DeclareMathOperator{\Li}{Li} \DeclareMathOperator{\supp}{supp} \emergencystretch=2em \setlist[enumerate]{topsep=5pt,itemsep=4pt} \title[An unconditional Artin--Matthews theorem]{An Unconditional Artin--Matthews Theorem for Simultaneous Primitive Roots} \author{Vlado Vojdanovski} \date{} \subjclass[2020]{Primary 11N64; Secondary 11F30, 11N36, 11R45} \keywords{Simultaneous primitive roots, cubic characters, shifted convolution, sieve methods, prime predecessors} \hypersetup{pdftitle={An Unconditional Artin--Matthews Theorem for Simultaneous Primitive Roots}, pdfauthor={Vlado Vojdanovski}} \begin{document} \begin{abstract} Let $S$ be a finite set of integers different from $0$ and $\pm1$. We prove that the primes for which every member of $S$ is a primitive root have positive lower relative density precisely when neither of Matthews' two obstructions occurs. The proof uses the marked estimates for prime predecessors and uniform splitting estimates of OpenAI. Its principal step is a refinement of the large-prime-factor law for prime predecessors by a fixed cubic Frobenius condition. We obtain this refinement from a shifted bilinear estimate for cubic theta series, a divisor-weighted positive sieve estimate, and a prime-extraction argument that retains the natural Hecke coefficients. The resulting lower bound holds in every sufficiently large dyadic interval. \end{abstract} \maketitle \section{Introduction} For a nonempty finite set \[ S=\{a_1,\ldots,a_r\}\subset\Z\setminus\{0,1,-1\}, \] write $\mathcal P_S$ for the primes $p\nmid a_1\cdots a_r$ such that \[ \ord_p(a_i)=p-1\qquad(1\leq i\leq r). \] There are two elementary obstructions to the infinitude of this set: \begin{enumerate}[label=\textup{(\Alph*)},ref=\Alph*] \item\label{obs:A} The product of some odd-cardinality subset of $S$ belongs to $\Q^{\times2}$. \item\label{obs:B} The product of some even-cardinality subset of $S$ belongs to $-3\Q^{\times2}$, and, for every prime $\ell\equiv1\pmod3$, at least one element of $S$ is a cube modulo $\ell$. \end{enumerate} Here a cube modulo a prime includes zero. The subsets in these conditions consist of distinct elements of $S$. \begin{theorem}\label{thm:main} If neither \textup{(A)} nor \textup{(B)} holds, there are constants $c_S>0$ and $x_S$ such that \begin{equation}\label{eq:main} \#\{x0. \] If either \textup{(A)} or \textup{(B)} holds, $\mathcal P_S$ is finite. \end{theorem} Matthews established this obstruction classification under the generalized Riemann hypothesis~\cite{Matthews}. Anwar and Pappalardi proved its infinitude direction under Schinzel's Hypothesis~H and gave an elementary treatment of the finite obstructions~\cite{AnwarPappalardi}. For one integer, Bhargava has proved positive lower relative density by combining marked Type~II estimates with a uniform zero-free half-plane~\cite{Bhargava}. The conclusion here is likewise a lower-density statement; the exact conjectural density is not needed. The simultaneous problem has an additional cubic feature. Requiring every $a_i$ to be a quadratic nonresidue can force $p\equiv1\pmod3$. One must then ensure that none of the $a_i$ is a cube modulo $p$. This condition is encoded by Frobenius in \[ K=\Q(\zeta_3,a_1^{1/3},\ldots,a_r^{1/3}), \] and generally cannot be specified by rational congruences alone. We prove a large-prime-factor law with this fixed Frobenius condition. Once it is available, the remaining index obstructions are removed by uniform splitting estimates and an elementary product argument. \subsection{The analytic statement} Fix $0<\eps<1/4$ and a nonzero function $\Phi\in C_c^\infty((1,2))$ with $0\leq\Phi\leq1$. Put \begin{equation}\label{eq:giant-def} I_x=(x^{1-\eps},x^{1-\eps/2}],\qquad f_x(u)=\sum_{\substack{\ell\in I_x\\\ell\ {\rm prime}}}\ind_{\ell\mid u}, \qquad C_x=\sum_{\substack{\ell\in I_x\\\ell\ {\rm prime}}}\frac1\ell. \end{equation} On the support of $\Phi(u/x)$, the function $f_x$ takes only the values zero and one for sufficiently large $x$. Mertens' theorem gives \begin{equation}\label{eq:c-eps} C_x\longrightarrow c_\eps :=\log\frac{1-\eps/2}{1-\eps}>0. \end{equation} Let $F=\Q(\zeta_3)$ and $H=\Gal(K/F)$. Complex conjugation acts by inversion on $H$. Thus an inversion-stable subset $\mathcal C\subset H$ defines an unambiguous Frobenius condition at unramified rational primes that split in $F$. \begin{theorem}\label{thm:reservoir} Let $K=F(b_1^{1/3},\ldots,b_s^{1/3})$ for fixed nonzero rational numbers $b_i$, and let $\mathcal C\subset H$ be inversion-stable. Let $3\mid M$, and let $U\subset(\Z/M\Z)^\times$ consist of classes congruent to $1$ modulo $3$. Then \begin{align}\label{eq:reservoir} &\sum_{\substack{p\ {\rm prime}\\p\ {\rm unramified\ in}\ K}} \Phi\!\left(\frac{p-1}{x}\right) \ind_{p\bmod M\in U}\ind_{\Frob_p\in\mathcal C}f_x(p-1) \notag\\ &\hspace{25mm}= \left(\frac{|U|}{\varphi(M)}\frac{|\mathcal C|}{|H|} c_\eps\int_0^\infty\Phi(t)\,dt+o(1)\right)\frac{x}{\log x}. \end{align} Without a Frobenius condition, the same assertion holds for every fixed $M$ and every $U\subset(\Z/M\Z)^\times$, with $|\mathcal C|/|H|$ omitted. \end{theorem} The proof has three analytic parts. First we extend the interior prime-predecessor law to fixed arithmetic progressions. Next we prove that the nontrivial cubic trace terms have zero mean in the statistic \eqref{eq:giant-def}. This uses a weight-one extension of the shifted bilinear estimate of Assing, Blomer, and Li~\cite{ABL}, followed by a positive sieve bound with the divisor function as a majorant. Finally, finite character expansion gives Theorem~\ref{thm:reservoir}. \subsection{External inputs and notation} We use the marked Type~II coefficient criterion of \cite[Lemma~10.2]{AR}, the correlation and prime-extraction framework of \cite[Sections~3--7]{PD}, and the block sieve of \cite[Lemma~2.9]{S}. The precise forms used below are stated where they enter the argument. The Hecke zero-free input is \cite[Theorem~1.1 and Lemma~4.8]{QRH}: finite-order Hecke $L$-functions over $F$ have no zeros in $\Re s>7/8$, with conductor-uniform polynomial growth in a fixed strip. The splitting estimate used at the end is \cite[Lemmas~9.1--9.2 and proof of Proposition~2.2]{AR}. We write $e(t)=\exp(2\pi it)$, $L=\log x$, and $\tau(n)$ for the divisor function. The functions $\omega(n)$ and $\Omega(n)$ count prime factors without and with multiplicity, respectively, and $P^-(n)$ denotes the least prime factor, with $P^-(1)=\infty$. All implied constants may depend on the fixed field, characters, and smooth functions under discussion. Each modulus and each finite family of auxiliary intervals is fixed before $x$ tends to infinity. The order of the later auxiliary limits will be specified explicitly. \input{preliminaries} \input{progression} \input{hecke} \input{spectral} \input{sampling} \input{sieve} \input{arithmetic} \input{bibliography} \end{document}