  We study the minimal number of variables required by a totally positive
definite diagonal universal quadratic form over a real quadratic field $\mathbb
Q(\sqrt D)$ and obtain lower and upper bounds for it in terms of certain sums
of coefficients of the associated continued fraction. We also estimate such
sums in terms of $D$ and establish a link between continued fraction expansions
and special values of $L$-functions in the spirit of Kronecker's limit formula.
