Explicit Tauberian estimates for functions with positive coefficients

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Abstract

If f{hook}(x)=∑anxn has an≥0 for all n, then for each x>0 for which the series converges we have nn≤x-nf{hook}(x) for each n. By choosing that x which minimizes the upper bound one obtains a "saddle point estimate" for each an 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.

Original languageEnglish (US)
Pages (from-to)187-197
Number of pages11
JournalJournal of Computational and Applied Mathematics
Volume41
Issue number1-2
DOIs
StatePublished - Aug 20 1992

Keywords

  • Rankin's method
  • Tauberian theorem
  • generating function

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