Asymptotic behavior of some power series with \(\zeta\)-functions in the coefficients (Q688959)

The asymptotic behaviors of four closely related special cases of power series with \(\zeta\)-functions in the coefficients have appeared in the literature [\textit{S. D. Chowla} and \textit{D. Hawkins}, J. Indian Math. Soc., New Ser. 26, 115--124 (1962; Zbl 0112.30203); \textit{D. P. Verma}, Indian J. Math. 6, 121--127 (1964; Zbl 0147.06103); \textit{J. Tennenbaum}, Math. Scand. 41, 242--248 (1978; Zbl 0371.10029); \textit{D. P. Verma} and \textit{S. S. Prasad} [J. Bihar Math. Soc. 7, 37--40 (1983; Zbl 0573.10028)]. A function which involves a real parameter is introduced; the parameter can be specialized to produce all of these special cases. The asymptotic behavior is obtained for this function for all real values of the parameter.