$\epsilon_1\le\min{(\epsilon,\rho/3)}$ since $\omega(n)\to\infty$ as $n\to\infty$ arbitrarily slowly. Hence, noting that $u=\theta/2\pi$, we obtain that the required inequality holds uniformly for $\delta_n\le\mid\theta\mid<\pi$.

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We now recall (\[balpha\]) from Lemma 2. It implies that $$\mathcal{B}_b^{1/2}(e^{-\alpha_n})\sim
 (h(\rho+1))^{1/2}\alpha_n^{-1-\rho/2}, \quad n\to\infty.$$ Combining this asymptotic equivalence with the result of Lemma 4, we obtain Hayman “decay” condition \[7; p. 565\], namely, $$\label{decay}
 \mid f_b(e^{-\alpha_n+i\theta})\mid
 =o(f_b(e^{-\alpha_n})/\mathcal{B}_b^{1/2}(e^{-\alpha_n})), \quad
 n\to\infty,$$ uniformly for $\delta_n\le\mid\theta\mid<\pi$.

Eqs. (\[capture\]), (\[locality\]) and (\[decay\]) show that the function $f_b(x)$ is admissible in the sense of Hayman. Therefore, we can apply Thm. VIII.4 of \[7\] for its coefficients. We state this result in the next lemma.

Suppose that the weight sequence $b$ satisfies Meinardus conditions $(M_1)-(M_3)$. Then, the asymptotic for the total number of weighted partitions is given by $$\label{totalnr}
 p_b(n) \sim \frac{e^{n\alpha_n} f_b(e^{-\alpha_n})}{\sqrt{2\pi\mathcal{B}_b(e^{-\alpha_n})}}$$ as $n\to\infty$, where $\alpha_n$ is the unique solution of (\[eqhay\]) whose asymptotic expansion is given by (\[alphan\]) and $\mathcal{B}_b(e^{-\alpha_n})$ is defined by (\[balpha\]). $\rule{2mm}{2mm}$

[*Remark.*]{} The asymptotic equivalence (\[totalnr\]) is in fact Meinardus asymptotic formula \[13\] for the number of weighted partitions of $n$. Here we give the formula in a slightly different form, which is more convenient for our further asymptotic analysis. One can easily show the coincidence of (\[totalnr\]) with 