Geodesic
On the Rank of Universal Quadratic Forms over Real Quadratic Fields
Documenta mathematica, Tome 23 (2018), pp. 15-34.

Voir la notice de l'article provenant de la source Electronic Library of Mathematics

We study the minimal number of variables required by a totally positive definite diagonal universal quadratic form over a real quadratic field $\Bbb Q(\sqrt{D})$ 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.
Classification : 11E12, 11R11, 11A55
Keywords: universal quadratic form, real quadratic form, number field, continued fraction, additively indecomposable integer, Kronecker's limit formula 
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     author = {Blomer, Valentin and Kala, V{\'\i}t\v{e}zslav},
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Blomer, Valentin; Kala, Vítězslav. On the Rank of Universal Quadratic Forms over Real Quadratic Fields. Documenta mathematica, Tome 23 (2018), pp. 15-34. http://geodesic.mathdoc.fr/item/DOCMA_2018__23__a61/