Iwasawa invariants of even K‐groups of rings of integers in the Z2‐extension over real quadratic number fields

Abstract Let be a real quadratic number field, and let denote its cyclotomic ‐extension. For each integer , let be the unique intermediate field in such that . By studying the 2‐adic divisibility of Dirichlet ‐series at negative integers, we derive an asymptotic formula that determines the order of the 2‐primary part of even ‐groups of rings of integers of for sufficiently large . As a corollary, we determine their and invariants. We also establish a lower bound for beyond which this asymptotic formula holds. Our results have two main applications: (1) For , or with , we determine the structure of the 2‐primary tame kernels . (2) We explicitly determine the three Iwasawa invariants for a family of real quadratic number fields, whose discriminants have arbitrarily many prime divisors.