On an inequality of Ramanujan concerning the prime counting function

Abstract

We discuss on the sign of $\mathcal{R}_{\alpha }(x):=\pi(x)^{2}-\frac{\alpha x}{\log x}\pi(\frac{x}{\alpha })$ for x sufficiently large, and for various values of α>0. The case α=e refers to a result due to Ramanujan asserting that $\mathcal{R}_{e}(x)<0$. Related by this inequality, we obtain a conditional result that gives the number N>530.2 such that $\mathcal{R}_{e}(x)<0$ is valid for x>e ^N. Moreover, we show that under assumption of validity of the Riemann hypothesis, the inequality $\mathcal{R}_{e}(x)<0$ holds for x>138,766,146,692,471,228. Then, in various cases for α, we find numerical values of x _α in which $\mathcal{R}_{\alpha }(x)$ is strictly positive or negative for x≥x _α .