Understanding the Riemann Hypothesis: From the primes through Riemann's zeros and Suzuki's screw function to Mittermeier's prime-power checkpoints and the open infinite-tail problem

Taken one at a time, the prime numbers are irregularly spaced; taken in bulk, they obey the prime number theorem, a precise asymptotic law for their average density. The Riemann hypothesis governs the scale of the fluctuations that this law leaves open: for weighted prime counting it confines the error to square-root scale, up to logarithmic factors. This handbook develops the whole chain of ideas — from the primes and their distribution, through the Riemann zeta function and its nontrivial zeros, to Suzuki's screw-function formulation and Mittermeier's prime-power checkpoint approach. Every notion that is needed is introduced at the point where it is first required. At the centre of the discussion is Suzuki's equivalence between the Riemann hypothesis and the nonnegativity of the associated real function Ψ = −gζ. Within this criterion, Mittermeier's analysis uncovers an exact interval geometry. Between consecutive prime powers the relevant curve is strictly convex, and the transition to convexity is governed by the plastic constant. Each interval therefore has exactly one minimum; this mathematically determined interval minimum is the checkpoint. Convex-dual and event-based representations turn the continuum sign condition into a sequence of uniquely characterized interval minima, and interval arithmetic with directed rounding certifies their positivity rigorously throughout the finite range up to q = 10¹⁰. The handbook develops this route from the elementary foundations to the current proof frontier of the approach. It explains why a complete interval certificate controls more than pointwise grid sampling, why the remaining infinite tail depends on delicate signed arithmetic cancellation, and how, within this framework, the unresolved part reduces to a concrete uniform inequality for a smoothed von Mangoldt sum. That inequality remains open. The Riemann hypothesis is not proved; the purpose of the handbook is to make the classical starting question, the checkpoint geometry, the rigorous finite result and the remaining mathematical obstruction intelligible as a single coherent line of argument.