  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.
