Meinardus conditions was weakened. Meinardus’ approach is based on considering two generating series: $$D(s)=\sum_{k=1}^\infty b_k k^{-s}, \quad s=\sigma+iy, \label{dofes}$$ and $$G(z)=\sum_{k=1}^\infty b_k z^k,\quad \mid z\mid\le 1. \label{gfbk}$$ Below we give the Meinardus scheme of conditions. Throughout the paper by $\Re(z)$ and $\Im(z)$ we denote the real and imaginary part of the complex number $z$, respectively.

$(M_1)$ The Dirichlet series (\[dofes\]) converges in the half-plane $\sigma>\rho>0$ and there is a constant $C_0\in(0,1)$, such that the function $D(s)$ has an analytic continuation to the half-plane $\{s:\sigma\ge -C_0\}$ on which it is analytic except for the simple pole at $s=\rho$ with residue $A>0$.

$(M_2)$ There exists a constant $C_1>0$ such that $$D(s)=O(\mid y\mid^{C_1}),\quad \mid y\mid\to\infty,$$ uniformly for $\sigma\ge -C_0$.

$(M_3)$ There are constants $\epsilon>0$ and $C_2(=C_2(\epsilon)>0)$, such that the function $g(\tau)=G(e^{-\tau}), \tau=\alpha+2\pi iu$, $u$ real and $\alpha>0$ (see (\[gfbk\])) satisfies $$\Re(g(\tau))-g(\alpha)\le-C_2\alpha^{-\epsilon},\quad \mid\arg(\tau)\mid>\pi/4,
 \quad 0\neq\mid u\mid\le 1/2,$$ for enough small values of $\alpha$.

The first assumption (M1) specifies the domain, say $\mathcal{H}$, in which $D(s)$ has an analytic continuation. The second is related to the asymptotic behavior of $D(s)$, whenever $|\Im(s)|\to\infty$. Functions, which are bounded by $O(|\Im(s)|^q), 0<q<\infty$, in certain domain, as $|\Im(s)|\to\infty$, are called functions of finite order. It is known that the 