Results on real zeros and the determinant: the prime bias, the interpolation theorem, the weighted determinant, the Frobenius divisibility and the theorem, with 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.

• Uniform exclusion of Landau–Siegel zeros (OpenAI) — Theorem A. There is an absolute constant c > 0 such that every real zero beta in (0,1) of every primitive nonprincipal real Dirichlet L-function of conductor q >= 3 satisfies (1 - beta) log q >= c. Proved in the last lesson.
• The logarithmic derivative of zeta — Lemma 1.1. -zeta'/zeta(s) < 1/(s-1) + 1 for every real s > 1.
• The prime bias (OpenAI) — Proposition 1.3. If beta is a real zero of L(s,chi), chi primitive real of conductor q, then the primes p <= X with chi(p) = 1 have sum of (log p)/p at most e(log q + delta (log X)^2/log q), delta = (1 - beta) log q.
• Mass of the primes with chi(p) = -1 — Corollary 1.4. The primes H < p <= X, p not dividing 2q, with chi(p) = -1 have sum of (log p)/p at least log X - 4 log q - 3 delta (log X)^2/log q - log H - 2c_M.
• Interpolation on a projected integer box (OpenAI) — Theorem 2.5. If A: C^4 -> C^3 is surjective with kernel spanned by a vector with Q-linearly independent coordinates, polynomials with deg_{z_j} <= t_j take arbitrary values at the points An, n in {0,...,N-1}^4, once t_j >= 3(N-1) and prod (t_j - 3(N-1) + 1) > (4N-3)^4.
• Rectangular boxes of exponents (OpenAI) — Corollary 2.6. For 1 <= H <= N, the monomial rows with exponents alpha_1 <= 32 H^(2/3) N^(4/3) and alpha_2, alpha_3 <= 32 H^(-1/3) N^(4/3) span.
• The weighted determinant — Lemma 3.4. Rows chosen greedily by the weight alpha_1 + H alpha_2 + H alpha_3 give N^4 rows of weight at most 96 H^(2/3) N^(4/3) and a nonzero determinant in Z[sqrt d, sqrt 2].
• Exponent sums (OpenAI) — Lemma 3.5. S_1 >= c_0 M H^(2/3) U and S_2 <= C_0 M H^(-1/3) U, with c_0 = 1/(4 * 97^2) and C_0 = 192.
• The Frobenius congruence — Lemma 4.3. theta^p = g_p(theta) mod p Z[sqrt d, sqrt 2] for odd p not dividing q, with g_p(sqrt d) = (d/p) sqrt d and g_p(sqrt 2) = (2/p) sqrt 2.
• Divisibility at admissible primes (OpenAI) — Proposition 4.4. For p > H, p not dividing 2q, chi(p) = -1, the determinant lies in p^(E_p) Z[sqrt d, sqrt 2], E_p = sum of floor(alpha_1/p).
• A zero-free region without exception — Corollary B. For an absolute c_B > 0, no L(s,chi) with chi mod q vanishes in Re s >= 1 - c_B/log(q(|Im s| + 2)).

Reading order

Read the lessons in order. The first lesson is the only analytic one; the second (interpolation) uses only linear algebra and convex geometry and can be read on its own.