Understanding the Riemann Hypothesis: From the Primes and the Zeta Zeros to Suzuki's Screw Function: Prime-Power Checkpoints, Computer-Assisted Positivity Certificates, and Weil's Quadratic Form

This handbook introduces the Riemann hypothesis from first principles to readers new to number theory, and gives an account, with proofs or proof sketches, of what the author’s checkpoint series proves about it. It follows one question: how far can the number of primes below a bound X stray from its average law? The storyline is a chain of exact translations from the concrete to the abstract. Riemann’s explicit formula turns the error of prime counting into waves set by the zeros of the zeta function, and Suzuki’s criterion turns the hypothesis that all nontrivial zeros lie on one line into the statement that a single real curve Ψ never dips below zero. The series attacks this sign question in three ways. By geometry: between consecutive prime powers the curve is strictly convex, so each such interval has exactly one lowest point. By certified computation: interval arithmetic with directed rounding proves Ψ(log X) > 0 for 1 < X ≤ 10¹⁰ from the primes, and for 1 < X ≤ 2 · 10²⁹ from verified zeta zeros and an explicit zero-density estimate. By analysis of the infinite remainder: everything beyond these ranges reduces to one inequality per falling episode, whose margin is a value of Weil’s quadratic form; required at all but finitely many episodes, this inequality is equivalent to the hypothesis. A comparison with other approaches, including the zero-free half-planes announced in 2026, and an explicitly hypothetical epilogue on the physical limits of certification complete the account. Every central statement carries a status label. The Riemann hypothesis remains open.