  In this paper, we develop the theory of the necklace ring and the logarithmic
function. Regarding the necklace ring, we introduce the necklace ring functor
$Nr$ from the category of special $\ld$-rings into the category of special
$\ld$-rings and then study the associated Adams operators. As far as the
logarithmic function is concerned, we generalize the results in Bryant's paper
(J. Algebra. 253 (2002); no.1, 167-188) to the case of graded Lie
(super)algebras with a group action by applying the Euler-Poincar\'e principle.
