Large gaps between primes

Large gaps between consecutive primes. The average gap, and the theorem that for every C > 0 a positive proportion of the gaps p_(n+1) - p_n exceed C log p_n, which answers a question of Erdős and Prachar on the increases of p_n/n. Smooth divisor sums and their averages, with a prime factor through the Bombieri-Vinogradov theorem; the singular series and Gallagher's theorem on its average; correlations of smooth divisor sums by Euler products near the pole of the zeta function; a square of signed divisor sums whose prime mass in an adjacent interval is controlled by neighbouring dimensions; and an alternating choice of test functions that makes this mass arbitrarily small.

Six lessons with proofs, examples and exercises with solutions. Written and self-checked by Claude Opus 5.5 (Anthropic); independent review is not complete.

1. Prime gaps and adjacent intervals
2. Divisor sums and primes in progressions
3. The singular series and its average
4. Correlations of smooth divisor sums
5. A square with little prime mass in the next interval
6. Cancellation between adjacent dimensions

Theorems and assumptions · Reader and editable sources · Course record

Prerequisites

• The Bombieri–Vinogradov theorem · Dirichlet L-functions and primes in progressions
• Dirichlet series and Euler products · The Riemann zeta function
• Counting primes by elementary means: Chebyshev and Mertens · The Riemann zeta function
• The prime number theorem · The Riemann zeta function
• Fourier transforms, finite spectra and convex separation
• Fourier inversion on the Schwartz space (the prerequisite bridges of the course on elliptic operators) and Fubini's theorem (core course Measure and Integration).