establish Hayman “locality” condition, which implies the asymptotic behavior of $f_b(x)$ in a suitable neighborhood of $x=1$.

Suppose that the weight sequence $b$ satisfies conditions $(M_1)$ and $(M_2)$ and $\alpha_n$ is the solution of (\[eqhay\]) given by (\[alphan\]). Let $$\label{delta}
 \delta_n=\alpha_n^{1+\rho/3}/\omega(n), \quad n\ge 1,$$ where $\omega(n)\to\infty$ as $n\to\infty$ arbitrarily slowly. Then $$\label{locality}
 e^{-i\theta
 n}\frac{f_b(e^{-\alpha_n+i\theta})}{f_b(e^{-\alpha_n})}
 =e^{-\theta^2 \mathcal{B}_b(e^{-\alpha_n})/2}(1+O(1/\omega^3(n))$$ uniformly for $\mid\theta\mid\le\delta_n$.

[*Proof.*]{} Applying Lemma 1, we observe that $$\begin{aligned}
\label{malko}
& & e^{-i\theta
 n}\frac{f_b(e^{-\alpha_n+i\theta})}{f_b(e^{-\alpha_n})}
 \\
 & & =\exp{\left(\frac{h}{\rho}((\alpha_n-i\theta)^{-\rho}
 -\alpha_n^{-\rho})-D(0)\log{\left(1-\frac{i\theta}{\alpha_n}\right)}
 -i\theta n +O(\alpha_n^{C_0})\right)}, \nonumber
\ \end{aligned}$$ where $h$ is given by (\[consth\]). Expanding $(\alpha_n-i\theta)^{-\rho}$ and $\log{(1-i\theta/\alpha_n)}$ by Taylor formula and using (\[aalpha\]), (\[balpha\]) and (\[eqhay\]), we obtain $$\begin{aligned}
 & & \frac{h}{\rho}((\alpha_n-i\theta)^{-\rho}-\alpha_n^{-\rho})
 -D(0)\log{\left(1-\frac{i\theta}{\alpha_n}\right)} -i\theta n
 \nonumber \\
  & & =i\theta(h\alpha_n^{-\rho-1} +D(0)\alpha_n^{-1} +D^\prime(0)-n)
 -\frac{\theta^2}{2} h(\rho+1)\alpha_n^{-\rho-2}
 -\frac{D(0)\theta^2}{2\alpha_n^2} \nonumber \\
 & & -i\theta D^\prime(0)
  +O(\mid\theta\mid^3\alpha_n^{-3-\rho})
 =i\theta(\mathcal{A}_b(e^{-\alpha_n}) -n +O(\alpha_n^{C_0-1}))
 \nonumber \\
 & & -\frac{\theta^2}{2}(\mathcal{B}_b(e^{-\alpha_n})
  +O(\alpha_n^{C_0-2}))
  +O(\mid\theta\mid) +O(\mid\theta\mid^3\alpha_n^{-3-\rho})
 \nonumber \\
 & & =-\frac{\theta^2}{2}\mathcal{B}_b(e^{-\alpha_n})
  +O(\delta_n\alpha_n^{C_0-1}) +O(\delta_n^2\alpha_n^{C_0-2})
 +O(\delta_n)+O(\delta_n^3\alpha_n^{-3-\rho}). \nonumber
 \end{aligned}$$ Substituting this into (\[malko\]) and taking into account (\[delta\]), we obtain (\[locality\]). $\rule{2mm}{2mm}$

To study the behavior of $f_b(e^{-\alpha_n+i\theta})$ outside the range $-\delta_n<\theta<\delta_n$ we need the Wiener-Ikehara Tauberian theorem on Dirichlet series. It tells us how condition $(M_1)$ implies an asymptotic estimate for the partial sums $\sum_{k=1}^n b_k$.

[**Wiener-Ikehara Theorem.**]{} [*(See 