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Prime numbers in typical continued fraction expansions.
Tanja I Schindler1, Roland Zweimüller1
1Fakultät für Mathematik, Universität Wien, Oskar-Morgenstern-Platz 1, 1090 Wien, Austria.
Bollettino Della Unione Matematica Italiana (2008)
|May 25, 2023
Summary
This study explores prime number occurrences in real number continued fractions using metrical number theory and ergodic theory. Probabilistic laws reveal patterns in these prime digit distributions.
Area of Science:
- Number Theory
- Ergodic Theory
- Probability Theory
Background:
- Continued fraction expansions are fundamental representations of real numbers.
- Understanding digit distribution in these expansions is a key problem in metrical number theory.
- Prime numbers exhibit unique properties that warrant investigation within number-theoretic frameworks.
Purpose of the Study:
- To investigate the probabilistic laws governing prime numbers appearing as digits in continued fraction expansions.
- To connect concepts from metrical number theory and ergodic theory to the distribution of prime digits.
- To analyze the frequency and patterns of prime numbers within these expansions.
Main Methods:
- Application of metrical number theory techniques.
- Utilizing tools from (infinite) ergodic theory.
- Probabilistic analysis of digit occurrences in continued fractions.
Main Results:
- Established probabilistic laws for prime number occurrences as digits.
- Demonstrated connections between number-theoretic properties and ergodic behavior.
- Quantified the distribution patterns of prime digits in continued fractions.
Conclusions:
- The study provides new insights into the distribution of prime numbers in number representations.
- Findings bridge number theory and ergodic theory through the analysis of continued fractions.
- Probabilistic laws offer a framework for understanding prime digit behavior.
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