Densities of 4-ranks of \(K_2({\mathcal O})\) (Q2781449)

All the possibilities of 4-rank of \(K_2 (O)\) are proved to be equal someway. Let \(r_i\) be the 4-ranks of \(K_2 (O_{F_i})\) for the quadratic number fields \(F_i={\mathbb Q} (\sqrt d_i)\), where \(d_i=p\ell , \;2p\ell , \;-p\ell , -2p\ell\) (for \(i=1,2,3,4\)) with primes \(p\equiv -1 \equiv -\ell\) (mod 8), \((\ell /p)=(p/\ell)=1\). We know each \(r_i\) takes two possible values (1 or 2 for \(i=1,2\); and 0 or 1 for \(i=3,4\)), and the tuple \((r_1,\;r_2, \;r_3,\;r_4)\) takes 8 possible values, from [\textit{P. E. Conner} and \textit{J. Hurrelbrink}, J. Number Theory 88, 263-282 (2001; Zbl 0985.11059)]. Fixing \(p\), here the author shows the densities of \(\ell\) for all possible values of \(r_i\) are equal.