Results on prime gaps and divisor-sum correlations, 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.

• The average gap — Proposition 1.1. (1/N) sum_(n<=N) (p_(n+1) - p_n)/log p_n -> 1, by the prime number theorem.
• At most a proportion 1/C of large gaps — Corollary 1.2. For every C > 0, the upper density of the n with p_(n+1) - p_n > C log p_n is at most 1/C.
• A positive proportion of large gaps — Theorem 2.1. For every C > 0 there are c(C) > 0 and N_0(C) with #{n <= N : p_(n+1) - p_n > C log p_n} >= c(C) N for all N >= N_0(C) (OpenAI, 2026).
• Increases of p_n/n — Corollary 2.2. The set of n with p_n/n < p_(n+1)/(n+1) has positive lower density (question of Erdős and Prachar).
• From adjacent intervals to gaps — Lemma 3.1. If for at least delta X integers m in (X, 2X] the interval (m, m+h] contains a prime and (m+h, m+2h] none, then at least delta X/h distinct primes in (X, 2X+h] are followed by a gap longer than h.
• A prime coordinate of a divisor sum — Lemma 1.3. If m + b_j is a prime with log(m+b_j)/log X larger than the budget, the smooth divisor sum equals the sum of one dimension less with the last argument of the test function set to 0.
• Bombieri-Vinogradov with moving endpoints — Proposition 3.1. For fixed sigma < 1/2, K >= 1, A > 0: sum_(q <= X^sigma) mu^2(q) K^omega(q) E_X(q) << X (log X)^(-A), where E_X(q) is the maximal error for primes in reduced classes in (X+a, 2X+a] with |a| <= D (log X)^B.
• Density sums — Proposition 4.2. The average over X < m <= 2X of a product of smooth divisor sums, optionally times theta(m+a_0), equals an explicit finite density sum up to O((log X)^(-A)), uniformly in shifts of size at most a power of log X; total budget below 1, or below 1/2 with the prime factor.
• Convergence of the singular series — Lemma 1.2. The singular series of a finite set converges absolutely, is nonnegative, vanishes exactly when the set meets every class modulo some prime, and equals 1 for a single point.
• Singular series of pairs — Lemma 1.3. S({0, delta}) = 0 for odd delta and 2 C_2 prod_(p | delta, p > 2) (p-1)/(p-2) for even delta, C_2 the twin-prime constant.
• Mean one modulo small primes — Corollary 2.2. The singular series truncated at y has average exactly 1 over all tuples of residues modulo the product of the primes up to y.
• Gallagher's average in boxes — Theorem 4.1. For s fixed and intervals I_1, ..., I_s of length h inside an interval of length K h, the sum of the singular series over distinct tuples with a_i in I_i is h^s (1 + O(1/log log h)).
• Correlations of smooth divisor sums — Theorem 1.1. Average of a product of smooth divisor sums, optionally times one shifted prime: L^(-t) (S(H) C + O(L^(-1/2) (log L)^A)), t the number of distinct divisor shifts, S the singular series, C an integral of complete mixed derivatives when every shift occurs at most twice.
• Fourier representation of test functions — Lemma 2.1. A smooth compactly supported function is an absolutely convergent Fourier integral of exponentials exp(-sum (1 + i u_l) v_l), with the same representation for its complete mixed derivative.
• Euler products and the singular series — Lemma 5.1. The non-zeta part of the Euler product of a density sum equals the singular series up to O(L^(-1/2) (log L)^A), uniformly in frequencies up to sqrt(L); valid when the singular series vanishes.
• Zeta near 1 — Lemma 6.1. For Re z > 0 and |z| <= 1, |zeta(1+z) - 1/z| <= 3.
• Cumulative test functions — Lemma 1.2. The cumulative integral of f in C_c^infty of the simplex {sum t_i < 1/8} is a test function whose signed complete mixed derivative is f; deleting a coordinate at 0 corresponds to f_j + T f_(j+1).
• Moments of the square — Proposition 2.1. For Z = sum over subsets S of the second block of D_(F_|S|)(m; S): E Z^2 -> w, E Z^4 = O(1), E Z^2 V_J -> lambda w, E Z^2 V_I -> v = lambda sum alpha_j ||f_j + T f_(j+1)||^2, and limsup E Z^2 V_J^2 <= (lambda + lambda^2 c_G^2) w.
• Deletion at a prime of the second block — Lemma 3.1. If m + a is prime with a in the second block, Z(m) is the same signed sum over subsets of the block without a, with F_j replaced by F_j + F_(j+1)(., 0).
• A profile spread over many scales — Lemma 1.1. For lambda, beta > 0 there is g >= 0 in C_c^infty((0, infinity)) with int g^2 = 1, int g = sqrt(lambda) and int u g^2 < beta.
• The cancelling family — Proposition 2.2. Alternating product functions on the r = floor(sqrt k) top levels give 1 - s^2/(k d^2) <= w <= 1 and v/lambda <= 2/r + (r/k)^2 + 4 s^2/(k d^2).
• Large gaps: the complete statement — Theorem 3.1. For every C > 0 a positive lower density of n satisfy p_(n+1) - p_n > C log p_n, and the increases of p_n/n have positive lower density (OpenAI, 2026).

Reading order

Read the first lesson for the theorem and its reduction to weights on adjacent intervals. The second, third and fourth lessons develop the analytic tools: averages of smooth divisor sums with the Bombieri-Vinogradov theorem, the singular series and Gallagher's average, and the Euler-product evaluation of divisor-sum correlations. The fifth lesson computes the moments of the weight, and the last makes the prime mass in the adjacent interval small and proves the theorem.