Publications and Research
Document Type
Article
Publication Date
Summer 7-29-2025
Abstract
The irrationality exponent of a real number α is defined by maximal value μ(α) ≥ 1 such that the inequality |p/q − α| < 1/qμ(α) is true for finitely many rational approxi- mations by relatively prime pairs p/q. The earliest result introduced the upper bound μ(π) ≤ 42 for the irrationality exponent of π. Subsequently, it was improved by many authors. The more recent result introduced the improved upper bound μ(π) ≤ 7.6063. This note introduces an improved irrationality exponent μ(π) ≤ 3. The optimal value is expected to be μ(π) = 2.
