of (\[realpartsfm\]) by the last $O$-estimate and applying inequality (\[sn\]) (see also (\[sinus\])) to its first term, for $\delta_n/2\pi\le\mid u\mid <1/2$, we obtain $$\Re{(F_{m,b}(e^{-\alpha_n+2\pi iu}))} -F_{m,b}(e^{-\alpha_n})
\le -C_3\alpha_n^{-\epsilon_1} +O(\log^\rho{\alpha_n^{-1}}).$$ Now, we are ready to compare the growth of (\[jtwo\]) with that of (\[joneas\]) whenever $m$ satisfies (\[mn\]). We have $$\begin{aligned}
\label{jtwoas}
& & \mid J_2(m,n)\mid \le\exp{(n\alpha_n+F_{m,b}(e^{-\alpha_n}))}
\nonumber \\
& & \times\int_{\frac{\delta_n}{2\pi}<\mid u\mid\le\frac{1}{2}}
\mid f_{m,b}(e^{-\alpha_n+2\pi iu})/f_{m,b}(e^{-\alpha_n})\mid du
\nonumber \\
& & =\exp{(n\alpha_n+ F_{m,b}(e^{-\alpha_n}))}
\int_{\frac{\delta_n}{2\pi}<\mid u\mid\le\frac{1}{2}}
(\Re{(F_{m,b}(e^{-\alpha_n+2\pi iu})} -F_{m,b}(e^{-\alpha_n})) du
\nonumber \\
& & =O(\exp{(n\alpha_n+F_{m,b}(e^{-\alpha_n})
-C_3\alpha_n^{-\epsilon_1} +O(\log^\rho{\alpha_n^{-1}}))})
\nonumber
\\
& & =O(e^{-C_3\alpha_n^{-\epsilon_1}}
\sqrt{2\pi\mathcal{B}_b(e^{-\alpha_n})}J_1(m,n)) =o(J_1(m,n)),\end{aligned}$$ where for the last $o$-estmate we have used (\[balpha\]). It is now clear that (\[sumint\]), (\[joneas\]) and (\[jtwoas\]) imply that $$p_b(n)\mathbb{P}(X_n\le m)
\sim\frac{e^{n\alpha_n}}{\sqrt{2\pi\mathcal{B}_b(e^{-\alpha_n})}}
e^{F_{m,b}(e^{-\alpha_n})}, \quad n\to\infty.$$ Subsequent application of the asymptotic equivalence (\[totalnr\]) from Lemma 5 implies that $$\label{dfas}
\mathbb{P}(X_n\le m) \sim
\exp{\{F_{m,b}(e^{-\alpha_n})-F_b(e^{-\alpha_n})\}},$$ where $\alpha_n$ and $m$ satisfy (\[alphan\]) and (\[mn\]), respectively.

Further on we shall study the asymptotic behavior of the exponent in (\[dfas\]). Our analysis will be based on a generalization of Perron formula that expresses the partial sums of a Dirichlet series as complex integrals of the inverse Mellin type transforms applied to the Dirichlet series itself. We shall use it in the form given 