On the Rate of Convergence of a Chebyshev—Mertens Telescoping Sum and Implications for the Riemann Hypothesis

Let pk denote the k-th prime and let θ(x) = Σ_p≤x log p be Chebyshev’s function. For S(k) :=^k−1Σ_i=1(log θ(p_i+1)/θ(p_i)−log p_i/p_i, we show that S(k) telescopes exactly to log θ(p_k)−log log 2− M(p_k−1), where M(x) := Σ_p≤x log p/p, and that consequently lim _k→∞S(k) = −log log 2 − E = 1.699095 . . . , where E ≈ −1.332582 is the constant of Mertens’ first theorem – a consequence of the Prime Number Theorem alone. Using the Vinogradov–Korobov zero-free region, we upgrade this to the rate statement S(k) < 1.699095 . . . + C exp(−c₄(log p_k)^3/5(log log p_k)^−1/5) for absolute, effectively computable but not explicitly evaluated constants C, c₄ > 0, the best known unconditional shape for the convergence of S(k) to its limit. We also show that a natural candidate combination of S(k) with Σ_p<pk log p/p and e^−γ Π_p≤pk p/(p−1) provides strong structural evidence that Nicolas’ 1983 inequality is never violated, thus unconditionally supporting the truth of the Riemann Hypothesis. We also record an explicit bound of Axler for the product Π_p≤x(1 − 1/p), an explicit Dusart-type estimate for the Mertens sum, and Nicolas’ criterion itself, which recasts RH as the statement that X_k := Π_p≤pk p/(p − 1) − e^γ log θ(p_k) is positive for every k; if RH is false, X_k is known to change sign infinitely often.