Robin’s Inequality at Colossally Abundant Numbers and Nicolas’ Inequality at Their Largest Prime Factor

Robin's criterion states that the Riemann Hypothesis holds if and only if \( σ(n) < eγ · n · loglog n \) for every \( n > 5040 \), and Nicolas' criterion states that it holds if and only if \( c(q) = log(q#/φ(q#)) - γ - loglogθ(q) > 0 \) for every prime \( q \), where \( q# \) is the primorial of \( q \) and \( θ \) is the Chebyshev function. Let \( N \) be a colossally abundant number, let \( ξ(N) \) be the real number associated with \( N \) by Nicolas, and let \( q \) be the largest prime not exceeding \( ξ(N) \). Writing \( Δ(N) = log(σ(N)/N) - γ - logloglog N \), we prove without any hypothesis that, whenever \( log q ≥ 31 \), \( 1.353/(√q log² q) ≤ Δ(N) - c(q) + 2√2/(√q log q) ≤ 3.970/(√q log² q) \). The range \( log q ≥ 31 \) contains the prime \( q \) of every colossally abundant counterexample, in accordance with the author's result that the largest prime factor of the least counterexample exceeds \( e31.018 \). Thus the functions \( Δ(N) \) and \( c(q) \) differ by \( 2√2/(√q log q) \) up to an explicit error of order \( 1/(√q log² q) \). This comparison is two-sided, but it is not an equivalence between Robin's inequality at \( N \) and Nicolas' inequality at \( q \): it does not decide Robin's inequality at \( N \) when \( c(q) √q log q \) lies between \( 2√2(1 - 1.4035/log q) \) and \( 2√2(1 - 0.4787/log q) \). As consequences, every colossally abundant counterexample to Robin's inequality yields a prime \( q \) with \( c(q) > 2.70/(√q log q) \); conversely, every prime \( q \) with \( log q ≥ 31 \) and \( c(q) > 2√2(1 - 0.4787/log q)/(√q log q) \) yields a colossally abundant counterexample, which gives a derivation of Robin's known theorem on colossally abundant counterexamples from Nicolas' oscillation theorem; and under the Riemann Hypothesis every colossally abundant number with \( log q ≥ 31 \) satisfies \( Δ(N) < -0.654/(√q log q) \), in accordance with the constant \( 2(√2-1) \) of Ramanujan's bound. For superabundant counterexamples, using the exponent bound of Nazardonyavi and Yakubovich, which gives \( log n ≥ ψ(pₖ) \) for the largest prime factor \( pₖ \), we prove \( c(pₖ) > 0.9991/(√{pₖ} log pₖ) \), a bound that applies to the least counterexample, which is superabundant, and we show that the least colossally abundant number not smaller than the least counterexample \( n \) is itself a counterexample \( N \) with \( log N - logξ(N) ≤ log n \).