TY - JOUR

T1 - Explicit Tauberian estimates for functions with positive coefficients

AU - Odlyzko, A. M.

PY - 1992/8/20

Y1 - 1992/8/20

N2 - 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.

AB - 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.

KW - Rankin's method

KW - Tauberian theorem

KW - generating function

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U2 - 10.1016/0377-0427(92)90248-V

DO - 10.1016/0377-0427(92)90248-V

M3 - Article

AN - SCOPUS:0040056875

VL - 41

SP - 187

EP - 197

JO - Journal of Computational and Applied Mathematics

JF - Journal of Computational and Applied Mathematics

SN - 0377-0427

IS - 1-2

ER -