Czechoslovak Mathematical Journal, Vol. 69, No. 2, pp. 443-452, 2019

A note on the distribution of angles associated to indefinite integral binary quadratic forms

Dragan Đokić

Received August 4, 2017.   Published online August 7, 2018.

Abstract:  To each indefinite integral binary quadratic form $Q$, we may associate the geodesic in $\mathbb{H}$ through the roots of quadratic equation $Q(x,1)$. In this paper we study the asymptotic distribution (as discriminant tends to infinity) of the angles between these geodesics and one fixed vertical geodesic which intersects all of them.
Keywords:  Weyl sum; indefinite integral binary quadratic form; real quadratic field; geodesic; asymptotic distribution
Classification MSC:  11L15, 62E20, 06B10
DOI:  10.21136/CMJ.2018.0370-17

[1] D. A. Buell: Binary Quadratic Forms. Classical Theory and Modern Computations. Springer, New York (1989). DOI 10.1007/978-1-4612-4542-1 | MR 1012948 | Zbl 0698.10013
[2] W. Duke, J. B. Friedlander, H. Iwaniec: Weyl sums for quadratic roots. Int. Math. Res. Not. 2012 (2012), 2493-2549; erratum ibid. 2012 (2012), 2646-2648. DOI 10.1093/imrn/rnr112 | MR 2926988 | Zbl 1300.11086
[3] H. Iwaniec, E. Kowalski: Analytic Number Theory. American Mathematical Society Colloquium Publications 53, American Mathematical Society, Providence (2004). DOI 10.1090/coll/053 | MR 2061214 | Zbl 1059.11001
[4] M. R. Murty: Problems in Analytic Number Theory. Graduate Texts in Mathematics 206, Springer, New York (2008). DOI 10.I007/978-1-4757-3441-6 | MR 1803093 | Zbl 1190.11001

Affiliations:   Dragan Đokić, Faculty of Mathematics, University of Belgrade, Studentski trg 16, 11000 Belgrade, Serbia, e-mail:

PDF available at: