On the rank of universal quadratic forms over real quadratic fields

Abstrakt

We study the minimal number of variables required by a totally positive definite diagonal universal quadratic form over a real quadratic field Q(root 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.

Klíčová slova

• universal quadratic form
• real quadratic number field
• continued fraction
• additively indecomposable integer