Geodesic
On the period of the continued fraction expansion for $\sqrt{d}$
Izvestiya. Mathematics , Tome 89 (2025) no. 1, pp. 26-49

Voir la notice de l'article provenant de la source Math-Net.Ru

If $d$ is not a perfect square, we define $T(d)$ as the length of the minimal period of the simple continued fraction expansion for $\sqrt{d}$. Otherwise, we put $T(d)=0$. In the recent paper (2024), F. Battistoni, L. Grenié and G. Molteni established (in particular) an upper bound for the second moment of $T(d)$ over the segment $x$. As a corollary, they derived a new upper estimate for the number of $d$ such that $T(d)>\alpha\sqrt{x}$. In this paper, we slightly improve this result of three authors.
Keywords: continued fraction, period of simple continued fraction expansion, trilinear Kloosterman sums. 
@article{IM2_2025_89_1_a2,
     author = {M. A. Korolev},
     title = {On the period of the continued fraction expansion for $\sqrt{d}$},
     journal = {Izvestiya. Mathematics },
     pages = {26--49},
     publisher = {mathdoc},
     volume = {89},
     number = {1},
     year = {2025},
     language = {en},
     url = {http://geodesic.mathdoc.fr/item/IM2_2025_89_1_a2/}
}
TY  - JOUR
AU  - M. A. Korolev
TI  - On the period of the continued fraction expansion for $\sqrt{d}$
JO  - Izvestiya. Mathematics 
PY  - 2025
SP  - 26
EP  - 49
VL  - 89
IS  - 1
PB  - mathdoc
UR  - http://geodesic.mathdoc.fr/item/IM2_2025_89_1_a2/
LA  - en
ID  - IM2_2025_89_1_a2
ER  - 
%0 Journal Article
%A M. A. Korolev
%T On the period of the continued fraction expansion for $\sqrt{d}$
%J Izvestiya. Mathematics 
%D 2025
%P 26-49
%V 89
%N 1
%I mathdoc
%U http://geodesic.mathdoc.fr/item/IM2_2025_89_1_a2/
%G en
%F IM2_2025_89_1_a2
M. A. Korolev. On the period of the continued fraction expansion for $\sqrt{d}$. Izvestiya. Mathematics , Tome 89 (2025) no. 1, pp. 26-49. http://geodesic.mathdoc.fr/item/IM2_2025_89_1_a2/