Explicit Tauberian estimates for functions with positive coefficients

If f{hook}(x)=∑a_nxⁿ has a_n≥0 for all n, then for each x>0 for which the series converges we have n_n≤x^-nf{hook}(x) for each n. By choosing that x which minimizes the upper bound one obtains a "saddle point estimate" for each a_n that has been known to be close to best possible in several cases. This paper presents a lower bound for summatory functions of the coefficients that is derived by elementary methods. It is not as sharp as the estimates that one obtains from most modern Tauberian theorems. However, this method can be used when Tauberian theorems are not applicable, for example, when one is dealing not with a single generating function but a sequence of them. Applications to partitions, integers without large prime factors, and other problems are presented.