Ramanujan’s Prime Counting Inequality Holds for All x ≥ 38 358 837 683

Let \( N_R \) denote the smallest positive integer such that Ramanujan’s inequality \( \pi(x)^2 < \frac{ex}{\log x}\,\pi(x/e) \) holds for every real \( x \geq N_R \). Assuming the Riemann hypothesis, Dudek and Platt showed that \( N_R \) = 38358837683. In this paper we prove the same value unconditionally. For this purpose we close the range \( e^{103} < x < e^{3158.442} \), which was not covered by previous unconditional results, using an explicit formula for \( ψ(x) \) together with new explicit information on zeros of the Riemann zeta function slightly above the height \( 3 \cdot 10^{12} \), up to which the Riemann hypothesis has been verified.