(\[limth\]) with $A=1$ and $\rho=1$ and $2$, respectively.

The method of our proof combines Hayman’s theorem for estimating coefficients of admissible power series \[11\] (see also \[7; Sect. VIII.5\]), a generalization of Perron formula, which yields the expression for partial sums of a Dirichlet series by a complex integral of the inverse Mellin transform applied to the Dirichlet series itself (see Thm 3.1 from the Supplement of \[18\]) and some Mellin transform computations.

We organize our paper as follows. Section 2 includes some auxiliary facts that we need further. Some proofs are omitted since they are given in \[10,13\]. In Section 3 we present the proof of Theorem 1. The Appendix contains some technical details related to the application of the generalized Perron formula \[18\].

Preliminary Results
===================

We start with a lemma establishing an asymptotic estimate for infinite product representation (\[euler\]) of the generating function $f_b(x)$. It has been proved by Meinardus \[13\] (see also \[2; Lemma 6.1\]).

Suppose that sequence $b$ is such that the associated Dirichlet series (\[dofes\]) satisfies conditions $(M_1)$ and $(M_2)$. If $\tau=\alpha+i\theta$, then $$f_b(e^{-\tau})
 =\exp{(A\Gamma(\rho)\zeta(\rho+1)\tau^{-\rho}-D(0)\log{\tau}
 +D^\prime(0)+O(\alpha^{C_0}))}$$ as $\alpha\to 0^+$ uniformly for $\mid\theta\mid\le\pi$ and $\mid\arg{\tau}\mid\le\pi/4$. $\rule{2mm}{2mm}$

Our first goal is to show that Meinardus’ conditions 