\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.