Results on Artin's primitive root conjecture and their assumptions

The course proves the statements below; each link leads to the statement with its assumptions and its proof. The lessons are self-checked by the writing AI; independent mathematical review is not complete.

• Primitive roots for every admissible base — Theorem 2.1. For every integer a other than -1 that is not a square there are c_a > 0 and x_a such that for x >= x_a at least c_a x/(log x)^2 primes p in (x, 2x) have p not dividing a and ord_p(a) = p - 1. Proved from Propositions 5.1 and 5.2 of lesson 1.
• The splitting test — Lemma 3.1. For a != 0 and primes p, q with p not dividing aq, p is unramified in K_q = Q(mu_q, a^(1/q)), and it splits completely there if and only if p = 1 mod q and a^((p-1)/q) = 1 mod p.
• A progression of quadratic nonresidues — Lemma 4.1. For admissible a and M = 8 times the product of the odd primes dividing a there are c in {2, 4} and u with (u, M) = 1, c | u - 1 and ((u - 1)/c, M/c) = 1 such that every prime p = u mod M satisfies p not dividing a and (a/p) = -1.
• Uniform complete splitting — Proposition 5.2. For |a| > 1, large x and large primes q <= exp((log x)^0.3), the primes p in (x, 2x] that split completely in K_q number O_a(x/(q(q - 1) log x) + x^(1 - 10^-6)), uniformly in q. Proved in lesson 15 from the zero-free strip.
• Simultaneous primitive roots — Theorem 6.1. For distinct positive primes q_1, ..., q_k, at least c x/(log x)^2 primes p in (x, 2x) have every q_i as a primitive root.
• Sextic reciprocity without supplementary factors — Proposition 2.1. For distinct exterior primes with allowed generators p, p' of a cyclotomic field containing mu_12, chi_p(p') = chi_p'(p); units enter only through the Hilbert pairing H_S.
• Kummer characters — Lemma 3.1. For an S-integer u, the ideal function a -> chi_a(u) agrees with a finite-order Hecke character kappa_u whose conductor has norm O(N rad(uR)); kappa_u = kappa_v exactly when u/v is a sixth power.
• Weil's index — Named result N2. The local Weil index depends only on the square class, satisfies w(a)w(b) = (a, b)_2 w(ab), is 1 at almost all places with product 1 over all places, and is a normalized quadratic Gauss sum at odd places. Used without proof (Rao; Liu and Zhu).
• The duplication identity — Proposition 5.1. At every exterior prime gamma_1 gamma_2 = -Lambda G, where Lambda comes from the Eisenstein normalization of cubic Jacobi sums and G from the Weil index; the normalized Gauss sums gamma_j are multiplicative up to the reciprocity twist.
• The reflected second moment — Proposition 4.1. The sum of |B_m(Z)|^2 over the row box is O(Z^(2M_0 - 1 + eps)), with the dyadic refinement Z^eps P^(1/2)(D + D^2 P/Z). Proved in lessons 6 to 10.
• Mellin continuation — Proposition 6.1. If H is holomorphic, bounded and nowhere zero on Re s >= 1 - delta_1 and the inverse Mellin integral of Z^(s - c) e^((s - s_0)^2) H(s)/L^S(s, eta) is O(Z^(1 - c - delta)), then L_F(s, eta) has no zero in Re s > 1 - delta.
• The cubic theta representation (Kazhdan and Patterson) — Named result N1. The cubic cover of PGL_2 over a number field containing mu_3 carries an exceptional theta representation with a nonzero global Whittaker functional, local uniqueness at the exterior primes and the stated local exceptional representations. Used without proof.
• Local Whittaker values — Proposition 4.1. W(a(p^e u)) = chi(u)^(2e) times Q^-k, Q^(-k-1/2) g_2 or 0 for e = 3k, 3k + 1, 3k + 2 with k >= 0, and 0 for e < 0, by an explicit shell computation.
• The exterior theta coefficient — Corollary 5.1. The product of the local Whittaker values at A = c n^3 with c squarefree is gamma_2(c)/(N c^(1/2) N n), and 0 when A has no such form.
• Exact Gaussian smoothing — Lemma 6.1. The Mellin transform of the archimedean profile V is C_0 A_0^s [Gamma(s + 5/6) Gamma(s + 7/6)]^r, and a rapidly decaying kernel with transform e^(s^2)/V^(s) turns V into the Gaussian weight P_G.
• Local reflection — Lemma 1.1. The local reflected sums equal s chi(t)^b with b = 8 - j at e = 0 and b = 4 + 2e for e > 0; for j = 1 only e = 0 survives, with a factor xi_p of modulus 1.
• The reflected expansion — Lemma 5.1. B_m(Z) is an absolutely convergent sum of frozen, row and column factors with the quadratic interaction chi_d(c')^3.
• A quadratic large sieve over number fields (Goldmakher and Louvel) — Named result N3. For a quadratic Hecke family, the sum over squarefree a with N a <= M of |sum over squarefree b with N b <= N of lambda_b chi_b(a)|^2 is O_eps((MN)^eps (M + N) sum |lambda_b|^2). Used without proof.
• The large sieve for the exterior symbols — Corollary 2.2. For complex z_c on squarefree exterior ideals of norm at most C, the sum over squarefree d of norm at most D of |sum z_c chi_d(c)^3|^2 is O_eps((C + D)(CD)^eps sum |z_c|^2).
• The probe bound (P1) — Proposition 4.1. I(Z^a, Z^b, Z) = O_eps(Z^(933/2000 + eps)).
• The Euler product of the probe — Proposition 5.1. F_u(xi, w, z) = zeta_F^S(6z) L^S(w, chi(u)) H_u(xi, w, z)/L^S(xi, eta conj chi(u)), with H_u holomorphic and O(q_u^eps), and H_1 bounded and nowhere zero.
• A large sieve for primitive characters — Lemma 2.1. For distinct primitive finite-order Hecke characters with conductor norms at most C_0 U, the mean square of sum b_n omega(n) over N n <= N is O((N + U^2)(NU)^eps sum |b_n|^2).
• Exceptional frequencies — Proposition 4.1. Outside a set of O(U^(1/2)) frequencies u with U <= q_u < 2U, 1/L^S(xi, eta conj chi(u)) = O(U^eps) for Re xi >= 0.998 and |Im xi| <= 2U^rho.
• The probe asymptotic (P2) — Theorem 4.1. I(Z^a, Z^b, Z) = c_0 f_eta(Z) + O(Z^(7/15 - 0.0004)).
• The zero-free strip — Corollary 4.2. For every cyclotomic field F containing mu_12 and every finite-order Hecke character eta of F, L_F(s, eta) has no zero in Re s > 1 - 1/20000. This is Theorem 5.3 of lesson 1.
• A uniform zero count (Hasanalizade, Shen and Wong) — Named result N4. |N_K(T) - (T/pi) log(d_K (T/(2 pi e))^(n_K))| <= 0.228 (log d_K + n_K log T) + 23.108 n_K + 4.520 for every number field K and T >= 1. Used without proof.
• Smooth prime ideal sums — Lemma 3.1. If zeta_K has no zero in Re s > 1 - delta, the smoothed prime ideal sum Theta_(K, phi)(x) is O(x + x^(1 - delta)(log D + n)), with a constant independent of K.
• A block sieve (Ford and Halberstam) — Lemma 2.1. If g(p) <= 1 - eta_0 and the sum of g(p) over v < p <= v^2 is at most C_0, the sifted weight is X prod (1 - g(p))(1 + O(e^-H)) + O(sum over d <= z^(4H + 2) of |E(d)|); with H = 2, an upper bound with moduli up to z^10.
• Buchstab's identity and an upper bound — Lemma 3.1. The integral from b to 1/2 of D_t(1 - t) dt/t equals D_b(1) - 1, and D_b(1) <= (e^-gamma/b)(1 + C_d e^(-1/(20b))) for small b.
• A one-sided marked Type II estimate — N5. Bilinear sums of the prime indicator minus its W-rough average against W-rough coefficients, weighted by a group-invariant F and the marked weight W(mn - 1), are O(X (log x)^(-D_*)) once the number K of prime groups is large. Proved as Theorem 2.3 of The marked Type II estimate, in the course Marked Type II estimates for prime predecessors (after OpenAI, The Poisson-Dirichlet law for prime predecessors, Theorem 3.1).
• The coefficient criterion — N6. The estimate N5 holds for every coefficient sequence on W-rough integers that cancels to (log x)^(-A) against characters of modulus at most (log x)^(A_0) twisted by m^(it), |t| <= 2X (log x)^B. Proved as Theorem 2.2 of The marked Type II estimate, in the course Marked Type II estimates for prime predecessors.
• Long prime polynomials and logarithmic phases — N7. Sums of chi(p) p^(-1 + iu) over primes in long intervals, and sums of n^(iu) over progressions, cancel at large heights u, uniformly in characters of modulus at most a power of log x. Proved as Theorem 1.1 of Long prime polynomials and Theorem 2.1 of A mean value theorem and logarithmic phases at large height, in the course Marked Type II estimates for prime predecessors.
• A rough-factor Type II estimate — Proposition 5.1. The marked Type II estimate holds for the indicator of x^gamma-rough integers minus D_gamma(log m/L)/(L V(W)) times the indicator of W-rough integers.
• Distribution of the weighted family — Lemma 4.1. The weighted family d = crQ + 1 is well distributed in progressions to moduli up to x^(1/2 - kappa/2), by Bombieri-Vinogradov, and its total weight X_0 is asymptotic to x J_0 A_Psi/(c phi(M/c) log x).
• Primes with controlled predecessors — Theorem 6.1. For M a positive multiple of 8, c in {2, 4} and u as in lesson 1, at least c_(M,c,u) x/(log x)^2 primes p in (x, 2x) satisfy p = u mod M and p - 1 = crQ with Q > x^0.9 prime and every prime factor of r in (exp((log x)^0.1), exp((log x)^0.3)). This is Proposition 5.1 of lesson 1.

Reading order

Nineteen lessons, to be read in order. The first states the theorem and reduces it to two propositions: many primes p = crQ + 1 in a progression of quadratic nonresidues, and a uniform bound for the primes that split completely in the Kummer fields. Lessons 2 to 14 prove a zero-free strip Re s > 1 - 10^-6 for the Hecke L-functions of cyclotomic fields containing the twelfth roots of unity: lessons 2 to 4 set up S-integers, sextic reciprocity, Kummer characters and the correction characters of cubic Gauss sums; lesson 5 defines the coefficient family and plans the argument; lessons 6 to 10 prove a reflected second moment with the cubic theta function, a Weyl reflection and a quadratic large sieve; lessons 11 to 14 evaluate a Poisson probe and its Euler product and detect the zeros. Lesson 15 deduces the uniform splitting bound, and lessons 16 to 19 construct the primes p = crQ + 1 with a block sieve, Bombieri-Vinogradov and marked bilinear estimates. The theory of Kazhdan and Patterson, Weil's index, the quadratic large sieve of Goldmakher and Louvel and a uniform zero count of Hasanalizade, Shen and Wong are used without proof, as the named results N1 to N4. The marked Type II estimates N5 to N7 of OpenAI's companion paper are proved in the course Marked Type II estimates for prime predecessors.