Results on marked Type II estimates for prime predecessors, 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.

• Twisted cancellation — Definition 2.1. A coefficient alpha on [H_m, 2H_m] has twisted cancellation if, for all fixed A_0, B, A, the sums (1/H_m) sum alpha_m chi(m) m^(it) are O(L^(-A)) for every character chi of modulus at most L^(A_0) and every |t| <= 2X L^B.
• The marked Type II estimate — Theorem 2.2. Let H_m, H_n >= x^delta with X = H_m H_n comparable to x, let alpha be supported on W-rough integers in [H_m, 2H_m] with |alpha| <= L^C and twisted cancellation, beta on W-rough integers in [H_n, 2H_n] with |beta| <= L^C, and F group-invariant with |F| <= 1. Then the sum of alpha_m beta_n F(mn - 1) W(mn - 1) is O(X L^(-D_*)) for K >= K_0 = 200(2D_* + 4C + 11) and every choice of band exponents. Proved in lessons 2 to 17.
• Prime predecessors — Theorem 2.3. The coefficient m^(iv)(1_prime(m) - 1_rough(m)/(V(W) log m)) on an interval of [H_m, 2H_m], |v| <= L^C, satisfies the hypotheses of Theorem 2.2; so the marked bilinear sums with this coefficient are O(X L^(-D_*)) for K >= K_0, uniformly in the interval and in v.
• Primes in progressions on short ranges — Lemma 1.1. For x^delta_1 <= N <= x^(1/delta_1), q <= L^(A_0) and (a, q) = 1, the primes p = a mod q in an interval of [N, 2N] number (1/phi(q)) times the integral of dy/log y over it, up to O(N L^(-A)); character sums over these primes have the principal-character main term only.
• Divisors in a progression — Lemma 2.2. For l, H' >= 1 with 1 + lH' <= Z, the sum over h <= H' of tau(1 + lh)^2 is at most 4H'(1 + log Z)^3 + 4Z^(3/4)(1 + log Z)^2.
• Well-spaced mean values — Lemma 3.2. For a Dirichlet polynomial of length Y and 1-separated points in an interval of length T, the sum of |P(t)|^2 is at most 2(T + 1 + 6 pi Y)(1 + log^2 Y) times the squared coefficient norm, with absolute constants.
• Rough integers twisted by characters — Lemma 4.2. For x^delta_1 <= H <= x^(1/delta_1) and characters chi of modulus at most L^(A_0), the sum of chi(n) over W-rough n in an interval of [H, 2H] is 1_(chi principal) V(W) times its length, up to O(H L^(-A)).
• Logarithmic phases with bounded height ratio — Theorem 3.1. For fixed Lambda >= 1, H >= 2, 0 <= u <= 1 and 1 <= |t| <= H^Lambda, every sum of (n + u)^(it) over an interval of [H, 2H] is O(H/|t| + H^(1 - c_Lambda)), with c_Lambda = 1/(d 2^(d+2)) and d = ceil(Lambda).
• Logarithmic phases on progressions — Corollary 4.1. For x^delta_1 <= N <= x^Lambda_1, q <= L^(C_0) and 1 <= |t| <= x^Lambda_1, the sum of n^(it) over n = a mod q in an interval of [N, 2N] is O((N/q)(1/|t| + x^(-c))), with c = c(delta_1, Lambda_1) > 0.
• A mean value theorem with uniform constants — Theorem 1.4. For every k >= 2 some s with k <= s <= C_1 k^3 satisfies J_(s,k)(M) <= exp(C_1 k^(C_2)) M^(2s - k(k+1)/2 + 1/100) for all M >= 1, with absolute constants C_1, C_2; proved by Linnik's p-adic iteration.
• Logarithmic phases at large height — Theorem 2.1. With T = log L, for exp(L/T^2) <= N <= 2x^5, q <= L^(C_0) and exp(L/(2T^2)) <= |v| <= 4x^3, every sum of n^(iv) over n = a mod q in an interval of [N, 2N] is at most (N/q) exp(-L/T^10).
• A zero-free strip at large height — Theorem 4.1. With T = log L, no Dirichlet L-function of modulus at most L^(C_0) vanishes in Re s >= 1 - cT^2/L, 1 <= |Im s| <= x^3, for some c = c(C_0) > 0; and |L'/L(s, chi)| = O(L) for 1 - cT^2/(2L) <= Re s <= 1 + 1/L, 2 <= |Im s| <= x^3/2.
• Long prime polynomials — Theorem 1.1. For x^tau/2 <= N <= x^(1/tau), characters of modulus at most L^(C_0) and L^(A+8) <= |t| <= x^2, the sum of chi(p) p^(-1+it) over the primes of any interval in [N, 2N] is O(L^(-A)).
• The prime coefficient has twisted cancellation — Proposition 2.1. For alpha_m = m^(iv)(1_prime(m) - 1_rough(m)/(V(W) log m)) on an interval of [H_m, 2H_m], (1/H_m) sum alpha_m chi(m) m^(it) = O(L^(-A)) for characters of modulus at most L^(A_0) and |t| <= 2X L^B, uniformly in the interval and in |v| <= L^C.
• The Cauchy reduction — Proposition 2.1. If the expanded squares Q_Y are O(XY L^(-D_1)) for every dyadic Y in range, the marked bilinear form is O(X L^(0.2 - D_1/2)).
• The determinant equation — Lemma 4.1. For coprime a, b and mn >= 2, there is an integer h with mn - 1 = ah and rs - 1 = bh if and only if bmn - ars = b - a.
• Two remaining estimates — Proposition 7.1. Theorem 2.2 of lesson 1 follows from bounds O(XY L^(-2D_* - 2)) for the minor part Q_Y^min and the major part Q_Y^maj of the determinant equation.
• Most ratios are good — Lemma 3.2. For fixed lists, the ratios r for which a state fails one of the two Diophantine tests (G1), (G2) form a set of measure at most exp(-c L^0.1).
• Stripping the pads — Corollary 5.2. The minor sum Q_Y^min equals the sum over the pad dyads d_0 of <f, T g>/d_0, up to O(XY L^(-A)) for every fixed A.
• Compression by arbitrary vectors — Lemma 1.1. For a positive semidefinite B and arbitrary vectors v_P, <sum c_P v_P, B sum c_Q v_Q> <= (sum_P <v_P, B v_P>) sum |c_P|^2.
• From the moment to the pairing — Proposition 3.1. If sum_P <u_P, (A A*)^R u_P> <= UV L^(-E_0 N), then |<f, A g>| <= C UV L^(4C - E_0) with an absolute constant C.
• An elementary Kloosterman bound — Lemma 1.2. |K_m(h, k)| <= C_eps m^(3/4 + eps) (h, k, m)^(1/4), from the fourth moment bound sum |K_m|^4 <= 2 tau_3(m) m^4 and the unit-scaling orbits.
• Equidistribution of root matrices — Theorem 2.1. For U, V >= x^delta and squarefree S <= exp(L^0.99), the primitive (u, v) with u/U, v/V and the ratio r in given intervals and root matrix congruent to g_0 mod S number UV|I_1||I_2||I_3|/(zeta(2)|SL_2(Z/S)|) + O(UV x^(-delta/200)).
• Bonferroni's inequalities for numbers — Lemma 3.1. For X_i in [0, 1], the product of the 1 - X_i differs from the alternating sum of the elementary symmetric functions e_0, ..., e_T by at most e_(T+1).
• Replacement of the root average — Theorem 4.1. The moment divided by UV equals 1/zeta(2) times the integral over the archimedean data of the model moment, with independent uniform kernel lines at the group primes, up to O(L^(-AN)) for every A.
• Orphan lifespans — Lemma 1.1. For a prime in no list, the expected damping factor is b'_p plus 1/(p + 1) times the sum over pairs of visits a < c with the same line of eta_a eta_c times the damping of the hits strictly between them.
• The operator identity — Theorem 4.1. The model moment divided by the product of the b'_p equals the sum of the histories of <b, G_0 E_0 G_1 ... E_(N-1) G_N b> with globally distinct births.
• The adjoint of an edge operation — Lemma 2.4. The adjoint of the piece of an edge that stores in the groups S and promotes in the groups I is the piece of the reversed edge that stores in I and promotes in S.
• Bounds for the ghost operations — Lemma 3.1. With the Schur weight 8^(memory size), every ghost operation has norm at most e^(9K) on memories of total size at most B = ceil(L^2).
• A balanced box — Lemma 3.1. If a block has a good source state, its positions have integer coordinates in a box with sides A', B', A'B' <= C_0 Y and Y^0.1 <= A', B' <= Y^0.9, in which the edge determinant is an ordinary 2 x 2 determinant.
• Clean edges contract — Theorem 5.1. For every fixed G, if A_0 >= G + 2, every clean edge piece has norm at most L^(-G).
• One omission tuple — Lemma 2.2. At a good source and outside an exceptional set of fresh draws, a promoted prime determines the omitted source labels and the other promoted primes, so the source symmetrization costs M^(-K).
• Contraction of every edge — Theorem 4.1. If A_0 >= G + 3 and 0.005 K > G + 2, every edge operation has norm at most L^(-G) on memories of total size at most B.
• The cycle identity — Lemma 1.1. The indicator that the performed values are distinct equals the sum over collections of disjoint blocks C of addresses of the products of (-1)^(|C| - 1)(|C| - 1)! times the indicators that each block is performed with equal values.
• The minor moment — Theorem 3.1. If A_0 >= E_0 + 7 and K >= 200(E_0 + 7), then sum_P <u_P, (A A*)^R u_P> <= UV L^(-E_0 N).
• The minor estimate — Theorem 4.2. With E_0 = D_1 + 4C + 2, A_0 >= E_0 + 7 and K >= 200(E_0 + 7), the minor part Q_Y^min of the determinant equation is O(XY L^(-D_1)).
• The major sum — Proposition 1.1. For every fixed D the major part Q_Y^maj is O(XY L^(-D)); the twisted cancellation of alpha is used here and only here.
• Sparse frequencies — Lemma 3.1. The unit intervals on which the polynomial of one isolated group prime exceeds L^(-A_s) number exp(O(L^0.91)), and the suprema of |N_beta|^2 over them add up to O(L^(2C + 1)).

Reading order

Seventeen lessons, to be read in order. The first states the theorem in a coefficient form and explains the plan. Lessons 2 to 6 supply the analytic inputs: primes and rough numbers, logarithmic phases at polynomial scales, a mean value theorem with uniform constants, a zero-free strip at large height, and long prime polynomials, which show that the prime coefficient has twisted cancellation. Lesson 7 removes the unknown function by Cauchy's inequality and reaches a determinant equation. Lessons 8 to 16 bound its minor-arc part: physical states and good positions, a high moment, equidistribution of root matrices and independent residue lines, lifespans and a memory of pending primes, truncation and adjoints, clean and dirty edges, and birth distinctness. Lesson 17 treats the major arcs and the final assembly. Every result used is proved in the course or in an earlier programme lesson, linked where it is used: primes, zero-free regions and exponential sums, and the analysis foundations (Cauchy's theorem, Hölder's inequality, Tonelli's theorem and Fourier inversion). Taylor's formula, the Chinese remainder theorem and convexity are taken from the core courses.