Evidence of equivalent conditions for the Riemann Hypothesis

This upload contains the corrected three-part Robin–Mertens proof package.

1. Appendix_RH_en.pdf
This appendix establishes the logarithmic bridge needed in the final Robin argument. It separates the primorial identity log n = θ(p_k) < p_k from the required statement for a least hypothetical Robin counterexample N>5040, namely p_k=P(N)<log N. The earlier swap-based interpretation is removed from the proof line; the remaining j_min=2 computation is retained only as a calibration for the delta-corrected beta envelope.

2. Appendix_RH_2_en.pdf
This appendix gives the corrected Mertens compensation step. Starting from an explicit Rosser–Schoenfeld/Mertens upper bound for β(N), it introduces a certified upper envelope B(N) for σ(N)/N. The positive Mertens remainder is absorbed directly by the envelope deficit β(N)-B(N). The certified J_1-tail is used only inside the abstract upper envelope B(N), not as a claim about the actual tail structure of SA/CA profiles.

3. RH_consolidated_2025.pdf
This consolidated manuscript combines the logarithmic bridge from Appendix RH with the certified-envelope compensation from Appendix RH 2. The final analytic chain is:
p_k=P(N)<log N,
σ(N)/N ≤ B(N),
and β(N)-B(N) absorbs the explicit Mertens surplus.
Together these steps yield Robin’s inequality for n>5040 and hence the Riemann Hypothesis.

This version supersedes earlier drafts in which the swap argument and the interpretation of the J_1-tail were stated too strongly. The corrected formulation removes those proof-line dependencies and uses only the minimal-counterexample bridge and the certified upper-envelope deficit.