  Consider exponential Carmichael function $\lambda^{(e)}$ such that
$\lambda^{(e)}$ is multiplicative and $\lambda^{(e)}(p^a) = \lambda(a)$, where
$\lambda$ is usual Carmichael function. We discuss the value of $\sum
\lambda^{(e)}(n)$, where $n$ runs over certain subsets of $[1,x]$, and provide
bounds on the error term, using analytic methods and especially estimates of
$\int_1^T \bigl| \zeta(\sigma+it) \bigr|^m dt$.
