sum of the Dirichlet series in (\[dofes\]) satisfies the finite order property in a closed half-plane contained in the half-plane of convergence $\sigma>\rho$ (see e.g. \[20; Sect. 9.3.3\]). The Meinardus second condition requires that the same holds for the analytic continuation of $D(s)$ in the whole domain $\mathcal{H}$. Finally, the Meinardus third condition implies a bound on $\Re(G(e^{-\tau}))$ (see (\[gfbk\])) for certain specific complex values of $\tau$. In some cases its verification is technically complicated. Since $$\label{regtau}
\Re(g(\tau))-g(\alpha) =-2\sum_{k=1}^\infty b_k
e^{-k\alpha}\sin^2{(\pi ku)}$$ and the inequality $\mid\arg(\tau)\mid>\pi/4$ implies that $\tan{(\mid\arg(\tau)\mid)}=2\pi\mid u\mid/\alpha>1$, condition $(M_3)$ can be also reformulated as follows: $$\label{sinus}
 S_n:=\sum_{k=1}^\infty b_k e^{-k\alpha}\sin^2{(\pi ku)}
 \ge C_2\alpha^{-\epsilon}, \quad 0<\frac{\alpha}{2\pi}<\mid
 u\mid\le 1/2,$$ for small enough $\alpha$ and some constants $C_2, \epsilon>0$ $(C_2=C_2(\epsilon))$ \[10; p.310\].

Moreover, Granovsky et al. \[10; Lemma 1\] proved that this inequality holds for any sequence $b_k, k\ge 1$, satisfying the inequality $b_k\ge Ck^{\nu-1}, k\ge
 k_0$, for some $k_0\ge 1$ and $C, \nu>0$. We notice that if $$\label{cknu}
 b_k=C k^{\nu-1}, \quad k\ge 1,$$ then $D(s)=C\zeta(s-\nu+1)$, where $\zeta$ denotes the Riemann zeta function. Therefore, $D(s)$ has a single pole at $s=\nu$ with residue $C>0$ and a meromorphic analytic continuation to the whole complex plane \[23; Sect. 13.13\]. These facts show that 