Sharp Approximations for the Ramanujan Constant

In this paper, the authors present sharp approximations in terms of sine function and polynomials for the so-called Ramanujan constant (or the Ramanujan $R$-function) $R(a)$, by showing some monotonicity, concavity and convexity properties of certain combinations defined in terms of $R(a)$, $\sin(\pi a)$ and polynomials. Some properties of the Riemann zeta function and its related special sums are presented, too.