Asymptotics of the Landau Constants and Their Relationship with Hypergeometric Functions

We examine the Landau constants defined by G(n) := Sigma(n)(m=0) 1/2(4m) ((m) (2m))(2) (n = 0, 1, 2, . . .) by making use of the celebrated Ramanujan formula expressing Gn in terms of the Clausenian (3)F(2) hypergeometric series. It is shown that it could be used to deduce other, mostly new, Ramanujan type formulas for the Landau constants involving the terminating and non-terminating hypergeometric series. In addition, by this approach we derive once again, in a simple and unified manner, almost all of the known results and also establish several new results for G(n). These new results include (for example) the generating function and asymptotic expansions and estimates for G(n)