the Meinardus original formula, applying the result of Lemma 1 to $f_b(e^{-\alpha_n})$ and replacing $\alpha_n$ and $\mathcal{B}_b(e^{-\alpha_n})$ by (\[alphan\]) and (\[balpha\]), respectively.

Further, we also need a bound on the rate of growth of the weights $b_k$, as $k\to\infty$. Using Wiener-Ikehara Tauberian theorem, Granovsky et al. \[10; p. 310\] showed that $b_k=o(k^\rho)$ as $k\to\infty$. We need this bound in a different slightly more precise form.

If the sequence of weights $b$ satisfies conditions $(M_1)$ and $(M_2)$, then there is a sequence of numbers $L_k, k\ge 1$, satisfying $\lim_{k\to\infty}L_k=0$ and such that $$\label{recur}
 b_k=(L_k-L_{k-1})k^\rho+(A+L_{k-1})k^{\rho-1}, \quad k\ge 2,
 \quad L_1=b_1-A,$$ where $A$ is the constant defined in condition $(M_1)$.

[*Proof.*]{} As in \[10; p. 310\], we rewrite (\[wi\]) in the following way: $$\frac{1}{k}\sum_{j=1}^k c_j
 =\frac{1}{k}c_k+\frac{1}{k}\sum_{j=1}^{k-1}c_j =c+L_k, \quad k\ge 2,$$ where $\lim_{k\to\infty}L_k=0$. We also set $L_1=c_1-c$. Then, for $k\ge 2$, we have $$\frac{1}{k}c_k=L_k-L_{k-1}+\frac{1}{k}(c+L_{k-1}).$$ To obtain (\[recur\]) it remains to set $c_k=k^{-\rho+1}b_k, k\ge 1,
 c=A$ and recall that the weights $b_k$ satisfy conditions $(M_1)$ and $(M_2)$ with $C_0, \rho>0$. $\quad\rule{2mm}{2mm}$

Now, we also recall formula (\[df\]) for the truncated products $f_{m,b}(x), m\ge 1$. Similarly to (\[capitalf\]), we set $$\label{capitalfm}
F_{m,b}(x) =\log{f_{m,b}(x)} =-\sum_{k=1}^m b_k\log{(1-x^k)}.$$ (Here we consider the main branch of the logarithmic function, assuming 