Robin’s Criterion on Divisibility (II)

Robin's criterion states that the Riemann hypothesis is true if and only if the inequality σ(n) < eγ · n · log log n holds for all natural numbers $n > 5040$, where σ(n) is the sum-of-divisors function of n and γ ≈ 0.57721 is the Euler-Mascheroni constant. We show that the Robin inequality is true for all natural numbers $n > 5040$ that are not divisible by some prime between $2$ and $1771559$. We prove that the Robin inequality holds when π²6 · loglog n' ≤ loglog n for some $n>5040$ where $n'$ is the square free kernel of the natural number n. The possible smallest counterexample $n > 5040$ of the Robin inequality implies that qₘ > e31.018189471, 1 < (1 + 1.2762log qₘ) · log(1.006479799241)log log n+ log Nₘlog n, (log n)^βₙ < 1.000208229291·log(Nₘ) and n < (1.006479799241)ᵐ · Nₘ, where Nₘ = ∏i = 1ᵐ qᵢ is the primorial number of order m, qₘ is the largest prime divisor of n and βₙ = ∏i = 1ᵐ qᵢ^aᵢ+1qᵢ^aᵢ+1-1 when n is an Hardy-Ramanujan integer of the form ∏ᵢ₌₁ᵐ qᵢ^aᵢ. By combining these results, we present a proof of the Riemann hypothesis. This work is an expansion and refinement of the article "Robin's criterion on divisibility", published in The Ramanujan Journal.