Variance of the additive weight deficit of equiprobable involution on the residue group

We find exact and asymptotic formulas for the variance of the random variable $\zeta_n$ which is equal to the weight deficit of random involution defined on the additive group of residues modulo natural number $n$. The asymptotic formula for $n\to\infty$ has the following form:
$$ \mathbf{D}{{\zeta}_{n}}=n\left(e^{-\frac{1}{2}}-\frac{3}{2}{{e}^{-1}} \right)\left(1+O\left(\frac{1}{n^{\frac{1}{3}}} \right) \right). $$