\def\Z{\mathbb{Z}} \def\Q{\mathbb{Q}} We give an essentially self-contained proof of the fact that a certain $p$-adic power series $$ \Psi= \Psi_p(T) \in T + T^{2}\Z[[T]]\;, $$ which trivializes the addition law of the formal group of Witt $p$-covectors $\widehat{\rm CW}_{\Z}$, is $p$-adically entire and assumes values in $\Z_p$ all over $\Q_p$. We also carefully examine its valuation and Newton polygons. We will recall and use the isomorphism between the Witt and hyperexponential groups over $\Z_p$, and the properties of $\Psi_p$, to show that, for any perfectoid field extension $(K,|\,|)$ of $(\Q_p,|\,|_p)$, and to a choice of a pseudo-uniformizer $\varpi = (\varpi^{(i)})_{i \geq 0}$ of $K^\flat$, we can associate a continuous additive character $\Psi_{\varpi}: \Q_p \to 1+K^{\circ \circ}$, and we will give a formula to calculate it. The character $\Psi_{\varpi}$ extends the map $x \mapsto \exp \pi x$, where $$\pi := \sum_{i\geq 0} \varpi^{(i)} p^i + \sum_{i<0} (\varpi^{(0)})^{p^{-i}} p^i \in K\;. $$ I will also present numerical computation of the first coefficients of $\Psi_p$, for small $p$, due to M. Candilera.