Year: 2026
Type: preprint
Abstract: In the thirties of the last century, I.\,M. Vinogradov proved that the inequality $\|pα\|\le p^{-1/5+\varepsilon}$ has infinitely many prime solutions $p$, where $\|\cdot\|$ 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 $\|pα\|\le p^{-1/3+\varepsilon}$ 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.
Source: arXiv (Cornell University)
Author: Molla, Esrafil Ali
Language: English
Subfield: Algebra and Number Theory
Field: Mathematics
Domain: Physical Sciences
Open Access status: green
OpenAlex ID: W7222927706