Geodesic
Estimates of the Levy constant for $\sqrt p$ and class number one criterion for $\mathbb Q(\sqrt p)$
Zapiski Nauchnykh Seminarov POMI, Analytical theory of numbers and theory of functions. Part 14, Tome 237 (1997), pp. 21-30 
E. P. Golubeva. Estimates of the Levy constant for $\sqrt p$ and class number one criterion for $\mathbb Q(\sqrt p)$. Zapiski Nauchnykh Seminarov POMI, Analytical theory of numbers and theory of functions. Part 14, Tome 237 (1997), pp. 21-30. http://geodesic.mathdoc.fr/item/ZNSL_1997_237_a2/
@article{ZNSL_1997_237_a2,
     author = {E. P. Golubeva},
     title = {Estimates of the {Levy} constant for $\sqrt p$ and class number one criterion for $\mathbb Q(\sqrt p)$},
     journal = {Zapiski Nauchnykh Seminarov POMI},
     pages = {21--30},
     year = {1997},
     volume = {237},
     language = {ru},
     url = {http://geodesic.mathdoc.fr/item/ZNSL_1997_237_a2/}
}
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Voir la notice du chapitre de livre provenant de la source Math-Net.Ru

Let $p\equiv3\!\pmod4$ be a prime, let $l(\sqrt p)$ be the length of the period of the expansion of $\sqrt p$ into a continued fraction, and let $h(4p)$ be the class number of the field $\mathbb Q(\sqrt p)$. Our main result is as follows. For $p>91$, $h(4p)=1$ if and only if $l(\sqrt p)>0.56\sqrt p\ L_{4p}(1)$, where $L_{4p}(1)$ is the corresponding Dirichlet series. The proof is based on studying linear relations between convergents of the expansion of $\sqrt p$ into a continued fraction.