Two algebraic number fields are said to be arithmetically equivalent if they share the same Dedekind zeta function. If this is the case, they must have certain invariants in common among which is the product $\,$ $hR$ $\,$ where $h$ is the class number and $\,$$R$$\,$ is the regulator.
Questions: 1) Let K and L be non-isomorphic but arithmetically equivalent number fields. On the face of it, the equality $\,$ $h(K)R(K)$ = $h(L)R(L)$ $\,$seems surprising. Does this really give an unexpected rational relation of dependence among the various logarithms involved or is there a simpler explanation if one passes to a suitable common extension field of K and L and expresses everything in terms of the information there?
2) Are there any examples of near misses where one zeta function is almost but not quite equal to another? $\,$ That is, if two Dedekind zeta functions (expanded as Dirichlet series) agree for all but finitely many terms, must they be identical?
Thanks very much.