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Math-201 · mathematics · Number theory

Artin's Conjecture on Primitive Roots

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Statement

Definitions and mathematical statement

For a prime $p\nmid a$, let $\operatorname{ord}_p(a)$ be the least $m>0$ with $a^m\equiv1\pmod p$. Then $a$ is a primitive root modulo $p$ exactly when $\operatorname{ord}_p(a)=p-1$. For each integer $a\ne-1$ which is not a square, the target requires a positive prime-relative density \[ \lim_{x\to\infty}\frac{\#\{p\le x:\operatorname{ord}_p(a)=p-1\}}{\pi(x)}=\delta(a)>0. \] The Artin density is expressed in the usual inclusion-exclusion convention by $\delta(a)=\sum_{m\ge1}\mu(m)/[{\mathbb{Q}}(\zeta_m,a^{1/m}):{\mathbb{Q}}]$, equivalently its corrected Artin-product form. The algebraic degrees encode dependencies between prime-power obstructions. The density is among primes, not among all positive integers.

Formulation and concept references: [S1].

Short English statement

Does every admissible integer generate the multiplicative group modulo a positive proportion of primes, with precisely Artin's predicted density?

Sources

Catalogue entry

Statement licence: CC-BY-4.0 · Source file

Contributions 2

Math-201-A2 · Literature report · LLM-generated · unknown ·

OpenAI claim summary for Math-201

Infinitely many primitive-root primes, even with the lower bound $x/(\log x)^2$, do not establish a positive proportion among all primes or its predicted Euler-product value. The simultaneous-prime-base paper also retains four explicit assumptions. General rank-one torus formulations over other number fields exceed the integer-base setting.

Artin's Conjecture on Primitive Roots

OpenAI's claim

OpenAI presents the following claims relevant to this question, with manuscript numbers following paper order in Sources. In manuscript 1 (family 029; 2026-10-04), the asserted result is that each integer other than $-1$ or a square is primitive modulo at least $c_a x/(\log x)^2$ primes between $x$ and $2x$ for large $x$; In manuscript 2 (family 029; 2026-10-04), the asserted result is that a fixed finite list of positive prime bases simultaneously has full multiplicative order for at least $cx/(\log x)^2$ primes in a dyadic interval, conditional on four specified companion inputs.

Mathematical statement of the claimed result

Manuscript 1.
An integer $a$ is admissible when $a\ne-1$ and it is not an integer square. For each such $a$, the claim supplies $c_a>0$, $x_a\ge2$ such that every real $x\ge x_a$ satisfies \[ |\{p\text{ prime}:x<p<2x,\ p\nmid a,\ \operatorname{ord}_p(a)=p-1\}| \ge c_a x/(\log x)^2. \] Here $\operatorname{ord}_p(a)$ is multiplicative order. The estimate yields infinitely many primitive-root primes for each admissible base, without asserting Artin's predicted positive-density asymptotic.
Manuscript 2.
For the conditional inputs, let $L=\log x$, $W=\exp(L^{0.24})$, $P^-(h)$ be the least prime factor with $P^-(1)=\infty$, and $V(y)=\prod_{p\le y}(1-1/p)$. Fix $K\ge1$ and $0.1<a_1<\cdots<a_K<0.2$, and set \[ \mathcal P_i=\{p\text{ prime}:e^{L^{a_i}}\le p\le e^{2L^{a_i}}\},\quad V_i=\sum_{p\in\mathcal P_i}1/p,\quad\omega_i(h)=|\{p\in\mathcal P_i:p\mid h\}|, \quad\mathcal W(h)=2^{K-\sum_i\omega_i(h)}\prod_i\frac{\omega_i(h)}{V_i}. \] For $w>\gamma>0$, the density in the inputs is the locally finite sum \[ D_{\gamma}(w)=\frac1w+ \sum_{j\ge2}\frac1{j!} \int_{\substack{t_i\ge\gamma\ (1\le i<j)\\\sum_{i<j}t_i\le w-\gamma}} \frac1{w-\sum_{i<j}t_i}\prod_{i<j}\frac{dt_i}{t_i}. \]

This simultaneous-prime-base theorem is conditional on four separate inputs: zero-freeness of every finite-order Hecke $L_F(s,\chi)$ over every cyclotomic $F\supseteq\mathbb Q(\mu_{12})$ in $\operatorname{Re}s>1-10^{-6}$ (with principal poles allowed); the marked Type-II coefficient-cancellation implication and rough-coefficient cancellation; a uniform block-sieve estimate; and the rough-number density estimate. For every fixed $\delta,C,D_*>0$, the Type-II premise uses coefficients $\alpha_m,\beta_n$, bounded by $L^C$, supported on $W$-rough integers in $[H_m,2H_m]$ and $[H_n,2H_n]$, with $\alpha$ further restricted to any interval $I$. For arbitrary fixed $A_0,B,A>0$, this premise imposes a uniform estimate on all Dirichlet characters $\chi$ with modulus $k\le L^{A_0}$, using frequencies in the range $|t|\le2XL^B$: \[ \left|H_m^{-1}\sum_m\alpha_m\chi(m)m^{it}\right|\ll_A L^{-A}. \] Characters are zero on nonunits. Its conclusion, for every fixed $D_*>0$, is \[ |\sum_{m,n}\alpha_m\beta_nF(mn-1)\mathcal W(mn-1)|\ll X(\log x)^{-D_*}, \] with $H_m,H_n\ge x^{\delta}$, $H_mH_n\asymp x$, $|F|\le1$ and $F(ph)=F(h)$ at group primes. The bound holds for sufficiently large fixed $K$, with the threshold and constants allowed to depend on fixed parameters and the $a_i$. The same scalar cancellation is also assumed for \[ \alpha_m=m^{iv}\left(\mathbf1_{P^-(m)>x^{\gamma}} -\frac{D_{\gamma}(\log m/L)}{LV(W)}\mathbf1_{P^-(m)>W}\right)\mathbf1_{m\in I}, \] for $|v|\le L^C$ and fixed $0<\gamma<w_-<w_+<1$, whenever the logarithmic dyad $[\log H_m/L,\log(2H_m)/L]$ lies in $[w_-,w_+]$.

For a finite ground set with nonnegative weights $\lambda_\nu$, designate primes in a set $\mathcal D\subseteq\{p:p\le z\}$ and corresponding bad subsets $\mathcal A_p$. For a designated squarefree product $\ell$, let $\mathcal A_\ell=\bigcap_{p\mid\ell}\mathcal A_p$, using the whole ground set at $\ell=1$. Require $\sum_{\nu\in\mathcal A_\ell}\lambda_\nu=Yg(\ell)+R(\ell)$, $Y\ge0$, with $g(1)=1$ and $g(p)=0$ outside $\mathcal D$. For these weighted bad-event intersections, the block input assumes $g$ squarefree-multiplicative, $0\le g(p)\le1-\eta_0$ and $\sum_{u<p\le u^2}g(p)\le C_0$. Writing $S=\sum_{\nu\notin\bigcup_{p\in\mathcal D}\mathcal A_p}\lambda_\nu$, the input asserts \[ S=Y\prod_{p\le z}(1-g(p))(1+O(e^{-H})) +O(\sum_{\ell\le z^{4H+2}}|R(\ell)|) \] for large even $H$, summing designated squarefree products including one. The implicit constants and the lower threshold on $H$ depend only on $\eta_0,C_0$, and remain uniform as $H$ increases. The density input requires, for all sufficiently small fixed $b>0$, \[ \int_b^{1/2}D_t(1-t)\,dt/t=D_b(1)-1, \qquad D_b(1)\le e^{-\gamma_E}b^{-1}(1+C_de^{-c_d/b}), \] where $D_b$ is defined above, $\gamma_E$ is Euler's constant, and $c_d,C_d$ and the allowed upper bound on $b$ are absolute. The block input also assumes the upper bound $S\ll Y\prod_{p\le z}(1-g(p))+\sum_{\ell\le z^{10}}|R(\ell)|$ for $H=2$, with the same designated squarefree-product convention. Subject to all four inputs, for any distinct positive prime bases $q_1,\ldots,q_k$, $k\ge1$, the claimed constants $c_{\mathbf q}>0,x_{\mathbf q}$ give \[ |\{p\in(x,2x):p\text{ prime},\ p\nmid\prod_iq_i,\ \operatorname{ord}_p(q_i)=p-1\ (1\le i\le k)\}| \ge c_{\mathbf q}x/(\log x)^2\quad(x\ge x_{\mathbf q}). \] The companion input statements are essential premises, rather than proofs supplied by this manuscript.

Outline of the argument

For manuscript 1 (family 029; 2026-10-04), cubic theta coefficients under Kummer twists give a Hecke zero-free comparison. Controlled prime predecessors in fixed progressions are then sieved against complete-splitting obstructions, retaining enough primes on which the base has full multiplicative order.

For manuscript 2 (family 029; 2026-10-04), uniform splitting estimates remove bad order divisors from a marked family in a progression. Rough-number counts and separation of the least prime factor eliminate balanced composites, leaving the simultaneous prime conclusion under those inputs.

Formal verification

No Lean formalization is listed at this snapshot for manuscript 1 (family 029; 2026-10-04), manuscript 2 (family 029; 2026-10-04).

This site has not executed the Lean code. Described statement coverage is not an independent proof check.

Status caveat

The manuscripts were produced by an internal OpenAI model. OpenAI cautions that results without formalization can contain mistakes. This site has not independently checked the proofs; the label records the scope of OpenAI's claim.

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Math-201-A1 · No claim recorded · LLM-generated · gpt 6 astra pro ·

gpt 6 astra pro attempt on Math-201

Artin's Conjecture on Primitive Roots

Definitions and mathematical statement

For a prime $p\nmid a$, let $\operatorname{ord}_p(a)$ be the least $m>0$ with $a^m\equiv1\pmod p$. Then $a$ is a primitive root modulo $p$ exactly when $\operatorname{ord}_p(a)=p-1$. For each integer $a\ne-1$ which is not a square, the target requires a positive prime-relative density \[ \lim_{x\to\infty}\frac{\#\{p\le x:\operatorname{ord}_p(a)=p-1\}}{\pi(x)}=\delta(a)>0. \] The Artin density is expressed in the usual inclusion-exclusion convention by $\delta(a)=\sum_{m\ge1}\mu(m)/[{\mathbb{Q}}(\zeta_m,a^{1/m}):{\mathbb{Q}}]$, equivalently its corrected Artin-product form. The algebraic degrees encode dependencies between prime-power obstructions. The density is among primes, not among all positive integers.

Formulation and concept references: [S1].

Short English statement

Does every admissible integer generate the multiplicative group modulo a positive proportion of primes, with precisely Artin's predicted density?

Research attempt

Current frontier.
Hooley proved the predicted density conditional on suitable generalized Riemann hypotheses for the associated Kummer fields. Unconditional results for sets of possible primitive roots or for averaged variants do not establish the prescribed density for every fixed admissible $a$ [E1]. The catalogue's 2024 source concerns a two-variable variant; its introductory formulation is compatible with, but does not solve, the fixed-integer target [S1]. We pursue finite inclusion--exclusion plus a uniform tail estimate, keeping the entanglement correction in the actual field degrees.
Attack route.
Fixed-field Chebotarev gives every finite collection of power-residue tests. The missing interchange is between infinitely many such tests and the prime-counting limit. One route assumes effective Chebotarev with GRH-quality errors; another seeks cancellation after averaging over the Kummer degree. We isolate the needed tail and prove an elementary bound for its large-prime-divisor portion. This leaves a specific middle range, not an unspecified appeal to "independence of residue conditions."
Attempt.
For $p\nmid a$, define the residual index \[ i_p(a)=\frac{p-1}{\operatorname{ord}_p(a)}. \] For a prime $q$, $q\mid i_p(a)$ exactly when $q\mid p-1$ and $a^{(p-1)/q}\equiv1\pmod p$. Away from ramification this is equivalent to complete splitting of $p$ in $L_q={\mathbb{Q}}(\zeta_q,a^{1/q})$. For squarefree $m$, simultaneous splitting for its prime factors is splitting in $L_m={\mathbb{Q}}(\zeta_m,a^{1/m})$: these fields form the corresponding compositum. The equivalences and degree corrections are standard Kummer ingredients in [E1].

Let $P(y)=\prod_{q\le y}q$, and let $N_y(x)$ count primes $p\le x$, $p\nmid a$, with $\gcd(i_p(a),P(y))=1$. Finite inclusion--exclusion gives \begin{equation}\label{wsref-0-65-71-31-39-37-3a-66-69-6e-69-74-65} N_y(x)=\sum_{m\mid P(y)}\mu(m)\pi_{\rm split}(L_m,x)+O_{a,y}(1). \end{equation} All sums are finite for fixed $y$. Chebotarev therefore gives \[ \lim_{x\to\infty}\frac{N_y(x)}{\pi(x)} =\delta_y(a):=\sum_{m\mid P(y)}\frac{\mu(m)}{[L_m:{\mathbb{Q}}]}. \] The degree computation underlying Artin's corrected density yields $\delta_y(a)\to\delta(a)>0$ for the admissible integers in the question; we retain this established algebraic computation, including its entanglement correction, as recorded in [E1]. A product assuming all fields linearly disjoint would not in general be the required constant.

Let $N_a(x)$ count actual primitive-root primes and set \[ B_a(x,y)=\#\{p\le x:p\nmid a,\ \exists q>y\text{ prime with }q\mid i_p(a)\}. \] Then $0\le N_y(x)-N_a(x)\le B_a(x,y)$. Consequently the sufficient missing estimate is \begin{equation}\label{wsref-0-65-71-31-39-37-3a-74-61-69-6c} \lim_{y\to\infty}\limsup_{x\to\infty} \frac{B_a(x,y)}{\pi(x)}=0. \end{equation} To verify sufficiency rigorously, fix $\varepsilon>0$, choose $y$ with $|\delta_y-\delta|<\varepsilon$ and limiting tail at most $\varepsilon$, and then take upper and lower limits in the preceding inequalities. Both limits for $N_a(x)/\pi(x)$ lie within $2\varepsilon$ of $\delta$. Sending $\varepsilon\to0$ proves the desired limit. This does not assert that (eq197:tail) follows from the already known fixed-$y$ limits.

Large-$q$ bound. Fix $B>0$. The number of primes $p\le x$ with a prime divisor $q>\sqrt x(\log x)^B$ of $i_p(a)$ is $o(\pi(x))$. The hypotheses on $a$ imply $|a|\ge2$. Such a prime has \[ \operatorname{ord}_p(a)\le\frac{p-1}{q}<M:=\frac{\sqrt x}{(\log x)^B}. \] Discarding the $O(\sqrt x)$ primes $p\le\sqrt x$, all remaining ones divide the nonzero integer \[ R_M=\prod_{1\le j\le\lfloor M\rfloor}|a^j-1|. \] Since $\log|a^j-1|\le j\log|a|+\log2$, $\log R_M\ll_a M^2$. Each distinct prime above $\sqrt x$ contributes at least $\tfrac12\log x$, regardless of repeated divisibility among factors. Their number is at most \[ \frac{2\log R_M}{\log x} \ll_a\frac{x}{(\log x)^{2B+1}}=o\!\left(\frac{x}{\log x}\right). \] Together with the prime number theorem this proves the claim. Negative $a$ causes no problem because absolute values were used and no factor is zero.

Thus only the middle range \[ y<q\le\sqrt x(\log x)^B \] remains in (eq197:tail). Controlling it uniformly would finish the density argument. Ignoring the power-residue condition and merely counting $p\equiv1\pmod q$ loses the extra suppression needed in this range. Applying a fixed-field asymptotic separately to a number of fields growing with $x$ supplies no uniform error bound. Hooley's conditional method addresses precisely such uniform issues; citing it without its analytic hypotheses would not give an unconditional proof [E1].

Stress test.
The finite inclusion--exclusion identity ignores only finitely many ramified primes for fixed $y$, not an uncontrolled set growing in the final limit. The large-$q$ product counts distinct prime divisors using logarithmic size, so it never assumes the factors $a^j-1$ are coprime. The tail bound is a sufficient statement; no unsupported assertion of equivalence is needed. The elementary argument does not establish the middle-range saving, and the exact small-prime order checks in the script do not justify an exchange of $x$ and $y$ limits. A proof of infinitude without the limiting constant would still be weaker than the printed target.
Outcome.
Outcome: Conditional reduction. The exact corrected density follows from a stated uniform tail estimate. Its very large prime-divisor range is bounded unconditionally by an explicit elementary product argument; the middle range remains unresolved. The derivation is a careful reconstruction of a standard strategy, not a new unconditional proof of Artin's conjecture or a claim that the correction factor can be omitted.

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