PreprintCC BY 4.0arXiv
On the average of class numbers of indefinite binary quadratic forms
Abstract
We prove that the sum of class numbers of primitive indefinite binary quadratic forms with integer coefficients over positive nonsquare discriminants less than X is O(X3/2√(loglog X)/(log X)3/2). This improves the longstanding classical upper bound O(X3/2/log X) and implies the same upper bound for the corresponding sum over real quadratic fields. Our main input is a uniform upper bound for the number of small fundamental solutions to Pell's equation. In particular, we prove that, for almost all nonsquare positive integers D, the fundamental solution εD of t2−Du2=1 is larger than any fixed power of D.
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BibTeX
@misc{diao2026average,
title = {{On the average of class numbers of indefinite binary quadratic forms}},
author = {Yijie Diao},
howpublished = {arXiv},
year = {2026},
month = oct,
doi = {10.48550/arxiv.2610.28001},
eprint = {2610.28001},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2610.28001},
note = {CC BY 4.0},
}
RIS
TY - UNPB
TI - On the average of class numbers of indefinite binary quadratic forms
AU - Yijie Diao
PY - 2026
DA - 2026/10/20
PB - arXiv
DO - 10.48550/arxiv.2610.28001
UR - https://arxiv.org/abs/2610.28001
AB - We prove that the sum of class numbers of primitive indefinite binary quadratic forms with integer coefficients over positive nonsquare discriminants less than X is O(X^3/2√(loglog X)/(log X)^3/2). This improves the longstanding classical upper bound O(X^3/2/log X) and implies the same upper bound for the corresponding sum over real quadratic fields. Our main input is a uniform upper bound for the number of small fundamental solutions to Pell's equation. In particular, we prove that, for almost all nonsquare positive integers D, the fundamental solution ε_D of t^2−Du^2=1 is larger than any fixed power of D.
ER -