Prime Numbers in Typical Continued Fraction Expansions

Tanja I. Schindler, Roland Zweimüller

correctmedium confidence

Category: Not specified
Journal tier: Specialist/Solid
Processed: Sep 28, 2025, 12:56 AM
arXiv Links: Abstract ↗PDF ↗

Audit review

The paper proves the extremal limit for the maximum of prime-filtered continued-fraction digits via an inclusion–exclusion argument under CF-mixing and the tail law µG({a' ≥ K}) ∼ 1/(log 2 · K log K). The candidate solution proves the same limit using a rare-event Poisson/no-hitting approach for the Gauss map together with the same tail estimate from the prime number theorem. Both arguments are sound and compatible; they differ primarily in technique.

Referee report (LaTeX)

\textbf{Recommendation:} minor revisions

\textbf{Journal Tier:} specialist/solid

\textbf{Justification:}

The paper gives a correct and well-motivated analysis of prime occurrences in continued fraction digits, culminating in a precise extremal limit law for the maximum with a nonstandard normalization. Methods are standard for the field (CF-mixing, PNT-based tails, inclusion–exclusion) but used effectively in a focused, specialized setting. Minor clarifications would further aid readers in connecting the inclusion–exclusion scheme to familiar Poisson/hitting-time heuristics.