The distribution of αp modulo one in real quadratic function fields
Abstract
In the thirties of the last century, I.\,M. Vinogradov proved that the inequality |pα|≤ p−1/5+ε has infinitely many prime solutions p, where |·| denotes the distance to a nearest integer. This result has subsequently been improved by many authors. In particular, Vaughan (1978) replaced the exponent 1/5 by 1/4, using his celebrated identity for the von Mangoldt function and a refinement of Fourier analytic arguments. The current record is due to Matomäki (2009), who showed that the inequality |pα|≤ p−1/3+ε has infinitely many prime solutions. This exponent 1/3 is considered the limit of current technology. Baier and the author established an analogue of Matomäki's result for the function field k=Fq(T) and its imaginary quadratic extensions. Moreover, for real quadratic extensions of k of class number one, they obtained an analogue of Vaughan's result (exponent 1/4). In this paper, we establish the analogue of Matomäki's result for all real quadratic extensions of k without any restriction on the class number.
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BibTeX
@misc{molla2026distribution,
title = {{The distribution of $αp$ modulo one in real quadratic function fields}},
author = {Esrafil Ali Molla},
howpublished = {arXiv},
year = {2026},
month = oct,
doi = {10.48550/arxiv.2610.23742},
eprint = {2610.23742},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2610.23742},
}
RIS
TY - UNPB
TI - The distribution of αp modulo one in real quadratic function fields
AU - Esrafil Ali Molla
PY - 2026
DA - 2026/10/16
PB - arXiv
DO - 10.48550/arxiv.2610.23742
UR - https://arxiv.org/abs/2610.23742
AB - In the thirties of the last century, I.\,M. Vinogradov proved that the inequality |pα|≤ p^−1/5+ε has infinitely many prime solutions p, where |·| denotes the distance to a nearest integer. This result has subsequently been improved by many authors. In particular, Vaughan (1978) replaced the exponent 1/5 by 1/4, using his celebrated identity for the von Mangoldt function and a refinement of Fourier analytic arguments. The current record is due to Matomäki (2009), who showed that the inequality |pα|≤ p^−1/3+ε has infinitely many prime solutions. This exponent 1/3 is considered the limit of current technology. Baier and the author established an analogue of Matomäki's result for the function field k=Fq(T) and its imaginary quadratic extensions. Moreover, for real quadratic extensions of k of class number one, they obtained an analogue of Vaughan's result (exponent 1/4). In this paper, we establish the analogue of Matomäki's result for all real quadratic extensions of k without any restriction on the class number.
ER -