Some properties of the difference between the Ramanujan constant and beta function

The authors present the power series expansions of the function R(a)−B(a)R(a)−B(a) at a=0a=0 and at a=1/2a=1/2, show the monotonicity and convexity properties of certain familiar combinations defined in terms of polynomials and the difference between the so-called Ramanujan constant R(a)R(a) and the beta function B(a)≡B(a,1−a)B(a)≡B(a,1−a), and obtain asymptotically sharp lower and upper bounds for R(a)R(a) in terms of B(a)B(a) and polynomials. In addition, some properties of the Riemann zeta function ζ(n)ζ(n), n∈Nn∈N, and its related sums are derived.