= Symmetric cyclotomic unit
{title2=$c_n(a,b)=\zeta_n^{(b-a)/2}(1-\zeta_n^a)/(1-\zeta_n^b)$}
Half exponents mean the inverse of two in the odd-order root-of-unity group. For $p\nmid ab$, the ratio is a real global unit and is compatible under the norms in the p-power tower. Its <Coleman power series> is $(1+T)^{(b-a)/2}(1-(1+T)^a)/(1-(1+T)^b)$, with constant term $a/b$. Its <higher logarithmic derivatives> vanish at odd indices and equal $(a^k-b^k)B_k/k$ at even indices, using $B_1=-1/2$.
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