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 ) at a = 0 and at a = 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 ) and the beta function B ( a ) ≡ B ( a , 1 − a ) , and obtain asymptotically sharp lower and upper bounds for R ( a ) in terms of B ( a ) and polynomials. In addition, some properties of the Riemann zeta function ζ ( n ) , n ∈ N , and its related sums are derived. [ABSTRACT FROM AUTHOR]