Sect. 6.5\]). The required estimate now follows from (\[diff\]), (\[sone\]) and (\[stwo\]).$\quad
\rule{2mm}{2mm}$

Our last forthcoming lemma supplies us with integral representations for $F_b(e^{-\alpha})$ and $F_{m,b}(e^{-\alpha}),
\alpha>0$, using the Dirichlet series (\[dofes\]) and its partial sums $$\label{dmofes}
D_m(s)=\sum_{k=1}^m b_k k^{-s}, \quad s=\sigma+iy, \quad m\ge 1.$$ The proof is based on a Mellin transform technique and can be found in \[13\], \[10; Lemma 2(ii)\] and \[2; Sect. 6.2\].

For any $\alpha, \Delta>0$, we have $$\label{efem}
F_{m,b}(e^{-\alpha})= \frac{1}{2\pi
i}\int_{\rho+\Delta-i\infty}^{\rho+\Delta+i\infty}
\alpha^{-s}\Gamma(s)\zeta(s+1)D_m(s)ds$$ and $$\label{ef}
F_b(e^{-\alpha})= \frac{1}{2\pi i}
\int_{\rho+\Delta-i\infty}^{\rho+\Delta+i\infty}
\alpha^{-s}\Gamma(s)\zeta(s+1)D(s)ds,$$ where $D_m(s)$ and $D(s)$ are defined by (\[dmofes\]) and (\[dofes\]), respectively.$\quad \rule{2mm}{2mm}$

Proof of the Main Result
========================

We apply first the Cauchy coefficient formula to (\[df\]) using the circle $x=e^{-\alpha_n+i\theta}, \pi<\theta\le\pi$, as a contour of integration ($\alpha_n$ is determined by (\[alphan\])). We obtain $$p_b(n)\mathbb{P}(X_n\le m) =\frac{e^{n\alpha_n}}{2\pi}\int_{-\pi}^\pi
f_{m,b}(e^{-\alpha_n+i\theta}) e^{-i\theta n}d\theta.$$ Then, we break up the range of integration as follows: $$\label{sumint}
p_b(n)\mathbb{P}(X_n\le m) =J_1(m,n) +J_2(m,n),$$ where $$\begin{aligned}
 \label{jone}
 & & J_1(m,n) =\frac{e^{n\alpha_n}}{2\pi}\int_{-\delta_n}^{\delta_n}
f_{m,b}(e^{-\alpha_n+i\theta}) e^{-i\theta n}d\theta \nonumber \\
& &=\frac{e^{n\alpha_n+F_{m,b}(e^{-\alpha_n})}}{2\pi}
\int_{-\delta_n}^{\delta_n}
\frac{f_{m,b}(e^{-\alpha_n+i\theta})}{f_{m,b}(e^{-\alpha_n})}e^{-i\theta
n}d\theta,\end{aligned}$$ $$\label{jtwo}
J_2(m,n)
=\frac{e^{n\alpha_n+F_{m,b}(e^{-\alpha_n})}}{2\pi}\int_{\delta_n<\mid\theta\mid\le\pi}
\frac{f_{m,b}(e^{-\alpha_n+i\theta})}{f_{m,b}(e^{-\alpha_n})}e^{-i\theta
n}d\theta$$ ($\delta_n$ and $F_{m,b}(x)$ are defined by (\[delta\]) and (\[capitalfm\]), respectively).

We start with an estimate for $J_1(m,n)$, expanding the integrand of (\[jone\]) by Taylor formula: $$\begin{aligned}
 & & 