Note on distribution of units of real quadratic number fields (Q1609938)

Let \(k\) be a real quadratic number field, \(p\) a rational prime split in \(k\) and \(P\) a prime ideal of \(k\) dividing \(p\). Let \(I_p\) be the index of the residue classes \(\bmod\;P\) containing units in the group \((Z_k/P)^*\) of restricted residue classes \(\bmod\;P\). It is stated that under GRH the density \(c_a\) of the set of primes \(p\) for which \(I_p\) equals a given positive integer \(a\), exists, and conditions are given for this density to be positive. Proofs will appear elsewhere.