I am studying Dirichlet characters and modular functions as part of my research, specifically working through concepts found in the paper "An explicit hybrid estimate for $L(1/2 + it)$" by Ghaith A. Hiary [ 1 ] (MR 3580112, Zbl 1401.11121, arXiv). In the course of my studies, in the proof of Lemma 3.3 of that paper I encountered the following identity that I am attempting to prove: $$ \chi(1 + C_1 x) = \prod_{j=1}^{\omega} \chi_j(1 + C_1(p_j^{a_j}) x)= \exp\left( \frac{2 \pi i \tilde{L} x}{D_1} \right), $$ The following conditions are given: $$ \begin{cases} \chi_j(1 + C_1(p_j^{a_j}) x) = \exp\left( \dfrac{2 \pi i \tilde{L}_j x}{D_1(p_j^{a_j})} \right),\\ \tilde{L} = q \displaystyle \sum_{j=1}^{\omega} \frac{\tilde{L}_j}{p_j^{a_j}}. \end{cases} $$ where
- $L_j$ is integer for all $j$,
- $ C_1(q) \cdot D_1(q) = q $,
- $ C_1(p_j^{a_j}) = p_j^{\lceil a_j / 2 \rceil} $,
- $ D_1(p_j^{a_j}) = p_j^{\lceil a_j / 2 \rceil - a_j} $,
- and finally both $ C_1 $ and $ D_1 $ are multiplicative functions.
We need to prove that $$ \exp\left( 2\pi i x C_1(q) \sum_{j=1}^{\omega} \frac{\tilde{L}_j}{p_j^{a_j}} \right) = \exp\left( 2\pi i x \sum_{j=1}^{\omega} \frac{\tilde{L}_j}{D_1(p_j^{a_j})} \right). $$
How can I manipulate this expression and prove the equality? Particularly, how do the multiplicative properties of $ C_1 $ and $ D_1 $ help in establishing this identity?
Any help or guidance would be appreciated!
