The remainder R(N) in the decomposition of ϑ(p_N): an exact identity, explicit bounds and typical size under the Riemann hypothesis .

Let p_N be the N-th prime and x̄ = li⁻¹(N). We study R(N) = ϑ(p_N) − ϑ(x̄) + (π(x̄) − N) ln x̄: an exact integral identity (R ≥ 0, sharp vanishing criterion), an explicit bound under the Riemann hypothesis for N ≥ 400, and, under RH, the typical size R(N) ~ ½ Ñ(x̄)² ln x̄/x̄ and p_N − x̄ ~ −Ñ(x̄) ln x̄ outside a set of logarithmic density at most 0.046, with limit laws under linear independence of the zeros. Exact computations up to x = 10^11 and at 10^10…10^15 agree. The proofs rely on cited results (two quoted from secondary sources) and have not been reviewed by a specialist; novelty is not established. Written with the assistance of Claude (Anthropic); the author is responsible for the content. © 2026 Sidi Mohamed Lakhal.