Let $X_1,\dots,X_n$ be independent real-valued random variables. Then, for every $u\in\mathbb R$,
\begin{align*}
\phi_{X_1+\cdots+X_n}(u)=\prod_{i=1}^n\phi_{X_i}(u).
\end{align*}
Probability & StatisticsProbability Theory
Discussion
The characteristic function of a sum of independent random variables factorises as the product of their individual characteristic functions.