Marked Type II estimates for prime predecessors

OpenAI's one-sided marked Type II estimate for prime predecessors: bilinear sums of the prime indicator minus its rough-number average against arbitrary rough coefficients, at the predecessors mn - 1 and weighted by marks from fixed bands of primes, are smaller than every power of log x. A coefficient criterion; a zero-free strip at large height and long prime polynomials; Cauchy's inequality and a determinant equation; physical states, root residues and independent lines; lifespans, a memory of pending primes and contraction on clean and dirty edges; birth distinctness; the major arcs by characters, Mellin separation and sparse frequencies.

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

1. The marked Type II estimate
2. Primes, divisors, mean squares and rough numbers
3. Logarithmic phases at polynomial scales
4. A mean value theorem and logarithmic phases at large height
5. A zero-free strip at large height
6. Long prime polynomials
7. Cauchy's inequality and the determinant equation
8. Physical states and good positions
9. From the moment to the pairing
10. Root residues
11. Independent residue lines
12. Lifespans and the memory space
13. Truncation, adjoints and absolute bounds
14. Clean edges and the lattice box
15. Dirty edges
16. Birth distinctness and the minor moment
17. The major term and the theorem

Theorems and assumptions · Reader and editable sources · Course record

Used in the course Artin's primitive root conjecture: the lesson Marked Type II estimates and rough factors applies Theorems 2.2 and 2.3 of the first lesson to replace a prime factor by a rough integer.

Prerequisites

• Lebesgue integration and smoothing — selected programme proofs (Distributions, kernels and analytic singularities)
• Schwartz functions and Fourier inversion (Distributions, kernels and analytic singularities)
• Zero-free regions and the exceptional zero · Dirichlet L-functions and primes in progressions
• The Siegel–Walfisz 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
• Entire functions of order one and the Hadamard product of ξ · The Riemann zeta function
• Exponential sums and a subconvex bound · The Riemann zeta function
• Nonvanishing on the line σ = 1 and the zero-free region · The Riemann zeta function
• The prime number theorem · The Riemann zeta function
• Growth bounds and wider zero-free regions · The Riemann zeta function
• Mean values of Dirichlet polynomials and of zeta on the critical line · The Riemann zeta function
• Measure and Hilbert space tools for Haar integration (Harmonic analysis on locally compact groups)
• Analytic number theory: Dirichlet characters and L-functions, the prime number theorem and the Siegel–Walfisz theorem, Mertens' estimates, exponential sums (Kusmin–Landau, van der Corput), Jensen's formula and Landau's lemma; Hilbert spaces and the Schur test; real and complex analysis: Taylor's formula, Hölder's inequality, Tonelli's theorem, Fourier inversion and Cauchy's theorem; the Chinese remainder theorem.