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
SN - 0377-0427
VL - 41
SP - 187
EP - 197
JO - Journal of Computational and Applied Mathematics
JF - Journal of Computational and Applied Mathematics
IS - 1-2
ER -