What type of study is this?

This is a Quantitative Study study.

October 8, 2025Open Access

Continued fractions with large prime partial quotients

Key Points

The study identifies the Lebesgue measure and Hausdorff dimension for sets with large prime partial quotients.
Findings include new asymptotics on the tail of the almost prime zeta function, enhancing previous research.
This extend existing theorems by Łuczak and Huang-Wu-Xu, demonstrating further complexity in continued fractions.
Results contribute to ongoing inquiries in number theory, with implications for understanding prime distributions.

Abstract

We determine the Lebesgue measure and Hausdorff dimension of various sets of real numbers with infinitely many partial quotients that are both large and prime, thus extending the well-known theorems by Łuczak (1997) and Huang-Wu-Xu (2020). To this end, we obtain new asymptotics on the tail end of the almost prime zeta function. Our results include some recent work by Schindler-Zweimüller (2023).

Continued fractions with large prime partial quotients

Key Points

Abstract

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