Variations on a theme of Apéry (2501.10090v2)

Abstract: Ap\'ery's remarkable discovery of rapidly converging continued fractions with small coefficients for $\zeta(2)$ and $\zeta(3)$ has led to a flurry of important activity in an incredible variety of different directions. Our purpose is to show that modifications of Ap\'ery's continued fractions can give interesting results including new rapidly convergent continued fractions for certain interesting constants.

• The paper introduces new continued fraction formulations by adapting Apéry's methods with innovative variable shifts and continuous generalizations.
• It validates these modifications through numerical examples, including identities involving gamma functions and unexpected simplifications.
• The work deepens understanding of rapid arithmetic approximations for ζ(2) and ζ(3), paving the way for further research in number theory.

Overview of "Variations on a theme of Apéry"

The paper "Variations on a theme of Apéry" by Henri Cohen and Wadim Zudilin revisits the profound contributions made by Apéry through rapidly converging continued fractions (CFs) for values such as $\zeta(2)$ and $\zeta(3)$. Although the original findings by Apéry sparked broad academic interest, this paper does not venture into uncharted territories of irrationality results. Instead, its primary ambition is to provide novel and elegant CF formulations by modifying Apéry's methods.

The paper begins by revisiting Apéry’s influential CFs, which are known for their impressive speeds of convergence. These include the formulations for $\log(2)$, $\zeta(2)$, and $\zeta(3)$, famously represented as "Tiny", "Small", and "Big Apéry”, respectively. Apéry's method, which leverages acceleration techniques, offers direct CF formulations, yielding precious arithmetic insights and rapid convergence.

Modifications of Apéry's Continued Fractions

A central theme of the paper is the modification of Apéry’s CFs. The authors explore two primary variants:

1. Variable Shifts: A shift of the index $n$ by a fractional amount (such as $n + 1/2$) is applied. This shift is meant to generate equivalent CFs.
2. Continuous Generalization: Introducing a variable $z$ in developing continuous generalization of the CFs, these transformations allow for new insights when evaluating the CFs for specific values of $z$.

Notably, the paper makes the argument that such modifications can be approached through computational tools and extensive CF databases. Nevertheless, recognizing the limits of new CFs remains challenging due to the inherent complexity of the transformations and the equivalency up to Möbius transformations.

Numerical Examples and Continued Fraction Results

Amid their investigation into these transformations, Cohen and Zudilin provide several specific numerical instances where modified CFs yield meaningful results. Noteworthy examples include:

• A modified "Small Apéry" CF achieved through fractional shifts, equating to specific identities involving gamma functions.
• Several transformations of CFs, which provide empirical validation through numerical identities and rare instances, noted for their serendipitous 'miracle' simplifications.

Theoretical Implications and Future Prospects

Though the paper's primary contribution lies in its introduction of new CFs, the implications for theoretical number theory and approximations of fundamental constants cannot be overstated. At a fundamental level, the methodologies could potentially inform future research on linear forms and irrationality measures. The paper's insights into transformations and variations offer deeper arithmetic comprehension for constants like $(\zeta(2))$ and $(\zeta(3))$.

Future directions may include developing systematic approaches for recognizing new CFs, exploring unexplored territories in gamma function CFs, and solidifying theoretical underpinnings related to shifts and Möbius transformations. Additionally, the integration of advanced symbolic computation for resolving CF challenges is a promising avenue, underscoring the synergy between theoretical math and computational tools.

The work by Cohen and Zudilin commands thoughtful consideration within the mathematical community, particularly for researchers invested in domains involving number theory, approximations, and continued fraction analysis. Despite the absence of new results in irrationality, the paper extends Apéry's foundational work, setting a stage for further advancements in mathematical approaches to fundamental constants.