LAUSR

Abstract Let l be a rational prime number. Assuming the Gross-Kuz'min conjecture along a Z l -extension K ∞ of a number field K, we show that there exist integers μ ˜ , λ ˜ and ν ˜ such that the exponent e ˜ n of the order l e ˜ n of the logarithmic class group C l ˜ n for the n-th layer K n of K ∞ is given by e ˜ n = μ ˜ l n + λ ˜ n + ν ˜ , for n big enough. We show some relations between the classical invariants μ and λ, and their logarithmic counterparts μ ˜ and λ ˜ for some class of Z l -extensions. Additionally, we provide numerical examples for the cyclotomic and the non-cyclotomic case.