The Riemann Hypothesis

Abstract

In mathematics, the Riemann Hypothesis is a conjecture that the Riemann zeta function has its zeros only at the negative even integers and complex numbers with real part 1 2. In 1915, Ramanujan proved that under the assumption of the Riemann Hypothesis, the inequality σ(n) < e γ ×n×log log n holds for all sufficiently large n, where σ(n) is the sum-of-divisors function and γ ≈ 0.57721 is the Euler-Mascheroni constant. In 1984, Guy Robin proved that the inequality is true for all n > 5040 if and only if the Riemann Hypothesis is true. In 2002, Lagarias proved that if the inequality σ(n) ≤ Hn + exp(Hn) × log Hn holds for all n ≥ 1, then the Riemann Hypothesis is true, where Hn is the n th harmonic number. We show certain properties of these both inequalities that leave us to a proof of the Riemann Hypothesis.